milling process

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 4, JULY 2012 901

Robust Active Chatter Control in the High-SpeedMilling Process

Niels J. M. van Dijk, Nathan van de Wouw, Ed J. J. Doppenberg, Han A. J. Oosterling, andHenk Nijmeijer, Fellow, IEEE

AbstractChatter is an instability phenomenon in machiningprocesses which limits productivity and results in inferior work-piece quality, noise and rapid tool wear. The increasing demandfor productivity in the manufacturing community motivates thedevelopment of an active control strategy to shape the chatter sta-bility boundary of manufacturing processes. In this work a controlmethodology for the high-speed milling process is developed thatalters the chatter stability boundary such that the area of chatter-free operating points is increased and a higher productivity can beattained. The methodology developed in this paper is based on arobust control approach using -synthesis. Hereto, the most im-portant process parameters (depth of cut and spindle speed) aretreated as uncertainties to guarantee the robust stability (i.e., nochatter) in an a priori specified range of these process parame-ters. Effectiveness of the proposed methodology is demonstratedby means of illustrative examples.

Index TermsActive control, delay systems, high-speed milling,machining chatter, magnetic bearings, robust controller synthesis.

NOMENCLATURE

Axial depth of cutmm.Constant for selecting controller input.Tooth passing frequencyHz.Actuator/tooltip forceN.Spindle-actuator transfer function matrix.Averaged cutting force matrixN/m .Controller outputA.Controller transfer function matrixA/m.Spindle speedrpm.Generalized plant.

Manuscript received February 02, 2011; revised April 17, 2011; acceptedApril 18, 2011. Manuscript received in final form May 14, 2011. Date of publi-cation June 20, 2011; date of current version May 22, 2012. Recommended byAssociate Editor R. Landers. This work was supported by the Dutch Ministry ofEconomic affairs within the framework of Innovation Oriented Research Pro-grammes (IOP) Precision Technology.N. J. M. van Dijk was with the Department of Mechanical Engineering, Eind-

hoven University of Technology, 5600 MB Eindhoven, The Netherlands. He isnow with the Philips Innovation Services, 5656 AE Eindhoven, The Nether-lands (e-mail: [email protected]).N. van de Wouw and H. Nijmeijer are with the Department of Mechanical

Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, TheNetherlands (e-mail: [email protected]; [email protected]).E. J. J. Doppenberg and J. A. J. Oosterling are with TNOScience and Industry,

2600 AD Delft, The Netherlands (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TCST.2011.2157160

Uncertainty model input/output.Generalized plant input/output vector.Actuator/cutter displacementsm.System/controller state vector.Periodic solution.Perturbations about periodic solution.Controller input signalm.Scalar uncertainty.Uncertainty set.Structured singular value.Delays.

I. INTRODUCTION

C HATTER is an instability phenomenon in machining pro-cesses which must be avoided at all times. The occurrenceof (regenerative) chatter results in an inferior workpiece qualitydue to heavy vibrations of the cutter. Moreover, a high level ofnoise is produced and the tool wears out rapidly. The occurrenceof chatter can be visualized in so-called stability lobes diagrams(SLD). In a SLD the chatter stability boundary between a stablecut (i.e., without chatter) and an unstable cut (i.e., with chatter)is visualized in terms of spindle speed and depth of cut.In the present day manufacturing industry, an increasing

demand for high-precision products at a high productivitylevel is seen. This motivates the desire for the design of ded-icated control strategies, which are able to actively alter thechatter stability boundary and therewith enable high materialremoval rates. Hereto, this paper presents an active controlstrategy which alters the stability lobes diagram in a selectivespindle speed range and, therewith, ensures a priori chatter-freemilling operations for a predefined domain of process param-eters (spindle speed and depth-of-cut) such that (chatter-free)operating points of higher material removal rate become fea-sible. Herein, an important challenge is to transform the modelof the high-speed milling process (which in general is describedby a set of nonlinear time-variant delay differential equations)into a generalized plant formulation making it suitable forrobust control design [1].Basically three methods exist in literature to control chatter.

The first method to avoid chatter is to adjust process parame-ters (i.e., spindle speed, feed per tooth, or chip load) such thata stable working point is chosen [2][4]. Although chatter can

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902 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 4, JULY 2012

be eliminated by adaptation of process parameters, the method-ology does not enlarge the domain of stable operation pointstowards those of higher productivity.A second method is to disturb the regenerative effect by con-

tinuous spindle speed modulation, see, e.g., [5], [6]. Althoughthe stability boundary is altered by spindle speed modulation[7], the method cannot be used in the case of high-speed milling.Namely, in order to disturb the regenerative effect, the spindlespeed variation should be extremely fast, while the speed ofvariation is limited by the inertia and actuation power of thespindle.The third method is to passively or actively alter the machine

dynamics to alter the chatter boundary. There are passive chattersuppression techniques that use dampers [8] or vibration ab-sorbers [9]. Passive dampers are relatively cheap and easy toimplement and never destabilize the system. However, the prac-tically achievable amount of damping is rather limited. More-over, vibration absorbers require accurate tuning of their naturalfrequencies and, consequently, lack robustness to changing ma-chining conditions.Active chatter control in milling has mainly been focused on

active damping of machine dynamics. In [10], active dampingof a milling spindle with piezoelectric actuators is demonstratedfor low spindle speeds. Kern et al. [11], [12] applied activedamping on a milling spindle equipped with an active magneticbearing (AMB). Minimization of the tooltip compliance using-synthesis, which will result in an increase of stable machiningpoints, is presented in [13]. An approach taking a different per-spective is presented in [14]. Herein, it is assumed that chatteroriginates from workpiece flexibilities. Active damping is ap-plied by using piezoelectric actuators and sensors, which aremounted to the thin-walled workpiece. In general one can saythat damping the machine or workpiece dynamics, either pas-sively or actively, results in a uniform increase of the stabilityboundary for all spindle speeds.All aforementioned active chatter control approaches aim

to attenuate chatter vibrations by applying damping to thespindle or the tool in an active way. In general, one can saythat damping the machine or workpiece dynamics, either pas-sively or actively, results in a uniform increase of the stabilityboundary for all spindle speeds. To enable more dedicatedshaping of the stability boundary (e.g., lifting the SLD lo-cally around a specific spindle speed), the regenerative effectshould be taken into account during chatter controller design.In [15], an optimal state feedback-observer controller withintegral control in the case of turning was designed taking theregenerative effect into account. The infinite-dimensionalregenerative delay term is written as a rational function viaPad approximation. Recently, Chen and Knospe [16] devel-oped three different chatter control strategies for the case ofturning: speed-independent control, speed-specified control,and speed-interval control. Moreover, it has been shown thatsignificant improvement in tailoring the stability lobes can beobtained using dedicated controllers, obtained via -synthesis,as compared to proportional-integral-differential (PID)-likecontrollers. Although the experimental setup discussed in [16]exhibits some aspects encountered in high-speed milling, acomprehensive active chatter control strategy tailored to the

full complexity of the HSM process is missing to this date.Moreover, except for the work in [11] and [12], all researchon active chatter control is limited to low spindle speeds (i.e.,below 5000 rpm).In this paper, an active chatter controller methodology for the

high-speed milling process is presented, which can guaranteechatter-free cutting operations in an a priori defined range ofprocess parameters such as spindle speed and depth of cut byemploying an active magnetic bearing as an actuator. Currentchatter control strategies for the milling process cannot providesuch a strong guarantee of a priori stability for a predefinedrange of working points. In general, the existing techniquesrequire a posteriori calculation of the set of stable workingpoints. The methodology developed in this paper is based on arobust control approach using -synthesis. Hereto, the most im-portant process parameters (depth of cut and spindle speed) aretreated as uncertainties. The proposed methodology will allowthe machinist to define a desired working range (in spindlespeed and depth of cut) and lift the SLD locally in a dedicatedfashion. In practice the maximum actuator force is limited.Hence, we propose a methodology for the robust stabilizationof high-speed milling operations while minimizing the controleffort. Effectiveness of the proposed control methodology isshown by means of an illustrative example.The paper is organized as follows. Section II presents a com-

prehensive model of the milling process. Moreover, stabilityproperties of the model will be discussed. Section III presentsthe problem statement of the active chatter control problem.Then, in Section IV, a comprehensive analysis is performed toselect an appropriate feedback signal for the active chatter con-troller input, such that the actuator forces needed for stabiliza-tion are significantly reduced. The model of the milling processas presented in Section II cannot be directly used in the ro-bust controller design procedure. Therefore, in Section V, somemodel simplifications will be discussed in order to construct amodel suitable for controller design. Section VI present the ro-bust control design procedure, based on a -synthesis approach.Results of the proposed strategy, when applied to an illustrativeexample, are presented in Section VII. Finally, conclusions aredrawn in Section VIII.

II. MILLING PROCESSThis section presents a comprehensive model of the milling

process which can be used to predict the occurrence of regener-ative chatter. Moreover, stability properties of the model willbe discussed. The model of the milling process as discussedin this section is originally presented in [17][20]. In Fig. 1,a schematic representation of the milling process is given. Ablock diagram of the milling process, with controller, is givenin Fig. 2. Each of the blocks in this figure will be explained inmore detail as follows.As can be seen from the block diagram in Fig. 2, the milling

process is a closed-loop position-driven process. The setpointof the milling process is the predefined motion of the tool withrespect to the workpiece, given in terms of the static chip thick-ness , where is the feed per toothand the rotation angle of the th tooth of the tool withrespect to the (normal) axis (see Fig. 1). However, the total

VAN DIJK et al.: ROBUST ACTIVE CHATTER CONTROL IN THE HIGH-SPEED MILLING PROCESS 903

Fig. 1. Schematic representation of the milling process.

Fig. 2. Block diagram of the milling process.

chip thickness also depends on the interaction between the cutterand the workpiece. Since, in general, the machine tool is notinfinitely stiff, the interaction between the cutter and the work-piece leads to cutter vibrations resulting in a dynamic displace-ment of the tool which is superimposed on the prede-fined tool motion. This results in a wavy surface on the work-piece. The next tooth encounters the wavy surface left behindby the previous tooth and generates its own waviness. This iscalled the regenerative effect and results in the block Delay inFig. 2. The difference between the current and previous wavysurface is denoted as the dynamic chip thickness

withthe delay. Hence, the total chip thickness encountered by tooth, , is the sum of the static and dynamic chip thickness:

. In the next sections, the compo-nents of the milling model will be described in more detail.

A. Cutting Force Model

The cutting force model (indicated by the Cutting block inFig. 2) relates the cutting forces acting at the tool tip of themachine spindle to the total chip thickness. The cutting forces in

tangential and radial direction for a single tooth are describedby the following exponential cutting force model:

(1)

where and are cutting parameterswhich depend on the workpiece material and is the axialdepth of cut. The function describes whether a toothis in or out of cut

else (2)

where and are the entry and exit angle of the cut, re-spectively. Via trigonometric functions, the cutting force caneasily be converted to (feed)- and (normal)-direction (seeFig. 1). Hence, cutting forces in - and -direction, and

, respectively, can be obtained by summing over all teeth,as shown in (3) at the bottom of the page, where

and

B. Spindle Dynamics and Actuator DynamicsThe cutting force interacts with the spindle rotor and tool

dynamics (block Spindle in Fig. 2). For the purpose of activechatter control, an actuator is implemented in the spindle rotor.The controller output is dictated to the actuator which, inturn, generates a force on the spindle.In general the spindle rotor and tool dynamics (jointly

called the spindle dynamics) can be modelled by a linearmulti-input-multi-output (MIMO) model. The model has fourinputs and four outputs. The inputs consist of the cutting forces

acting at the tool-tip in -/ -di-rection and the actuator forcesin -/ -direction induced at some point in the spindle, whichgenerally differs from the location at which the cutting forcesare acting (the tooltip). This leads to an inherent flexibilitybetween the actuator/sensor system and the cutting forces.The outputs of the spindle rotor dynamics model are thedisplacements of the tooltipand displacements measured at some position on thespindle, which are used for feedback. In this paper, the ma-chine spindle-toolholder-tool dynamics is modelled by twodecoupled subsystems (representing the dynamics in two

orthogonal directions perpendicular to the spindle axis)consisting of two mass-spring-damper systems to mimic theinherent compliance between actuator and tooltip, see Fig. 3,

(3)


Fig. 3. Schematic overview of spindle dynamics model, . :forces displacements at actuator, : forces displacements at tooltip.

TABLE IMILLING MODEL PARAMETERS

with masses , eigenfrequenciesand dimension-

less damping ratios. This is done in order to capture the inherent dynamics

between the actuator/sensor system (denoted by subscript )and the cutting tool (denoted by subscript ). The parametersof the machine spindle model and cutting force coefficients arelisted in Table I. Herein, the cutting model parameters (and ) are taken from [21] and spindle parameters are chosensuch that these represent realistic machine spindle dynamicsfor high-speed milling machines. The state-space equationsdescribing the rotor dynamic model are given as follows:

(4)

where is the state vector (the order of this modelprimarily depends on the order of the spindle-tool dynamicsmodel, which in this case equals ), ,

, and . The state-space ma-trices of the spindle rotor dynamic model are given in the ap-pendix. Next to the spindle rotor dynamics, an actuator modelis included. The active chatter control design procedure will bedeveloped for a model incorporating a nonlinear actuator modelfor an activemagnetic bearing (AMB). As described in the intro-duction an AMB is a common type of actuator applied to rotor

dynamic systems and in [11] feasibility of using such actuatorin the scope of high-speed milling has been shown. This moti-vates to pay special attention to this kind of actuator (model).The nonlinear model of an AMB driven in differential mode isgiven as follows [22]:

(5)

where are the specific AMB coefficients, is theso-called premagnetizing current (to compensate for gravity,etc.), the corresponding nominal gap displacement and

is the controller output (i.e., the input currents to theactuator) and the bearing displacements. In general, thedisplacements in the actuator journal are significantlysmaller than the gap width . In addition, the controller output

will be limited by the controller design methodology.Then, for control design the nonlinear model of the AMB maybe linearized about , which has already beensuccessfully performed for many applications as is describedin [22]. This results in the following linear model of the AMB:

(6)

where

(7)

(8)

C. Total Milling ModelIn the previous sections, the submodels for the cutting force

and spindle rotor, toolholder, tool, and actuator dynamics rep-resenting the different blocks in the milling model as given inFig. 2 are introduced. Substitution of the cutting force modeland actuator model, given in (3) and (6), respectively, into themodel of the spindle rotor, toolholder, and tool dynamics, givenin (4), yields the total milling model, shown in (9) at the bottomof the page. It can be seen that the model describing the millingprocess is set of nonlinear, time-dependent delay differentialequations (DDE). In the next section, the stability properties ofthe model will be analyzed.

D. Stability of the Milling ProcessIn this section, we will briefly address the stability analysis

exploited to determine chatter boundaries in the stability lobes

(9)


diagram. In the milling process the static chip thickness is peri-odic with period time . Here is the spindle speed inrevolutions per minute (rpm). In general, the uncontrolled (i.e.,

) milling model (9) has a periodic solution withperiod time [23]. To validate this fact let us adopt the followingdecomposition of :

(10)

where is a -periodic motion that can be considered asthe ideal motion when no chatter occurs and the perturba-tion term. When no chatter occurs, and the tool mo-tion is described by the following ordinary differential equation(ODE):

(11)

which follows from (9) by exploiting the fact that. The ODE in (11) is a linear system with a periodic

excitation with period time . Hence, when hasno eigenvalues at , for and all , thesolution exists, is unique and is -periodic [24]. There-fore, the periodic solution is (at least locally) asymptoticallystable when no chatter occurs and when chatter occurs it is un-stable. Therefore, the chatter stability boundary can be foundby studying the (local) stability of the periodic solution .To this end, the uncontrolled milling model is linearized aboutthe periodic solution which yields the following linearizeddynamics in terms of the perturbations :

(12)

where

(13)

As can be seen from (12) and (13), the linearized model is a de-layed, periodically time-varying system. Stability of these kindof systems can be assessed using, e.g., the semi-discretizationmethod of [25]. The main point of semi-discretization is thatonly the delay term is discretized, instead of the actual time-do-main terms. Unless stated differently, all stability lobes dia-grams, presented throughout this paper, are determined usingthe semi-discretization method.

III. PROBLEM STATEMENTRecall from the previous section that the SLD is determined

using the model which describes the perturbation vibrationsabout the (chatter-free) periodic solution of the milling process.Therefore, the controller design, as presented in this paper, willbe based on the model which is linearized about the periodicsolution with uncertainties in depth of cut and spindle speed

which results in an uncertainty in the delay .1 Moreover,chatter is defined as the loss of stability of this periodic solutionand stability of the milling process is based on the stability ofthe model describing the perturbations of the milling processaround the periodic solution. The aim of this paper is to designa finite-dimensional linear controller , which guarantees thefollowing: robust stability of the milling process (12) for the givenuncertainties in depth of cut , time delay ;

performance by minimizing the total amount of actuatorenergy needed to stabilize the uncertain milling process.

Hereby, it is assumed that the controller , with controllerinput and output current , has the followingstate-space description:

(14)

Herein, , , , ,and with the order of the controller. Thechoice of the controller input signal will be discussedin Section IV. The linearized uncertain model of the millingprocess, given by (12), (13), cannot be directly used in thestandard robust controller design procedure. Therefore, afterdiscussing the selection of the controller input signal , twomodel simplifications will be presented in Section V such thatthe infinite-dimensional time-varying model (12), (13), is trans-formed into a finite-dimensional linear time-invariant (LTI)model. In this way, the model can be used in a robust controldesign procedure, which will be presented in Section VI.

IV. CONTROLLER INPUT SIGNAL

An important part of any control system is the choice of thefeedback signal used for control.From the discussion in Section II-D it becomes clear that the

nominal (chatter-free) solution of the milling model is periodicwith period time . Moreover, chatter is defined as the loss ofstability of this periodic solution and stability of the millingprocess is based on the stability of the model describing the per-turbations of the milling process around the periodic solution.Then, two possibilities arise in selecting the feedback signalwhich serves as an input to the controller in (14), namelyas follows:1) full output feedback, i.e., the total (measured) displace-ments are used for feedback: in (14);

2) perturbation feedback, i.e., the perturbation (chatter) vibra-tions are used as feedback signal,where denotes the periodic solution of the nominalmodel given by (9): in (14).

In Section VI, the design of a linear dynamic output feedbackcontrol law characterized by the transfer function and witha state-space description as defined in (14), is pursued. Next, thecontroller input signal will be denoted as

(15)1Note that and are not uncertain in practice, but since we aim to stabilize

the milling process in a range of working points , we treat the parametersas uncertainties in the control design.


where is the periodic solution at the measuredoutput. Moreover, is a constant indicating whether full output

or perturbation feedback is applied.The implication of the choice for either one of the two con-

troller input signals will be demonstrated next. It can easily beshown that the stability properties of the closed loop, in case ofthe linear actuator model are the same for both choices of thefeedback signal [either , or in (15)].First, it will be shown that the chatter-free -periodic solu-

tion of the closed-loop system (9), (14); will bedifferent for both choices of the controller input signal. To showthis, consider the following decompositions of the state vectors

and :

(16)

Then, let us combine (9) with (14) and substituteand . Using the fact that is -periodic (i.e.,

), this results in the following closed-loopdynamics:

(17)

It can be seen that the closed-loop dynamics in (17) governingthe periodic solution is an LTI system with a -periodic distur-bance since both and are periodic with . Thenit can be concluded that when full output feedback is applied(i.e., ), with the assumption that has no eigenvalues at

, for and all , the solutionexists, is unique and is -periodic [24]. Note that this solutiondiffers from the periodic solution of the open-loop dynamicsgiven in (11). The conditions imposed on the eigenvalues ofwill in general be satisfied, since the controller has to renderthe closed-loop system stable also for (i.e., the eigen-values of will lie in the open left half plane). However,when and consequently perturbation feedback is ap-plied, the eigenvalues of are given by the eigenvalues of

and . Then, there exists a unique, -periodic,solution when has no eigenvaluesat , for and has no eigen-values with real part equal to zero. The conditions imposed on

will typically be satisfied since, in general, theAMB actuator is designed such that the decrease in stiffness,due to the negative stiffness effect of an electromagnetic actu-ator, see [22], is significantly smaller than the stiffness of thespindle rotor. Consequently, under such conditions, in case ofperturbation feedback, is the only solution of (17) sat-isfying . Therewith, the periodic solution

of (17) becomes equal to the solution of the open-loop pe-riodic solution of (11). Hence, in case of perturbation feedback

(i.e., ) the periodic solution will beequal to that of the uncontrolled system (and , i.e.,the nominal control action is zero).On the other hand, for full output feedback (i.e., )the periodic solution will be different from that of the un-controlled system (and will in general be non-zero, i.e.,the steady-state control action

does not vanish).Second, it can be shown that for both choices of the feed-

back signal the linearization of (9) with (14) aboutis given by

(18)

with as defined in (13). Clearly, for both choices of thecontrol input signal, the resulting SLD will be the same (sincethe perturbation dynamics (18) does not depend on the constant). Moreover, in the case of perturbation feedback thenominal control action vanishes in steady-state, which is not thecase for full output feedback . As a result, the choicefor perturbation feedback is favorable from the point of view ofbounding the control action.From the analysis, presented above, it can be concluded that

the SLD with active chatter controller (14) does not depend onthe chosen controller input signal. This is due to the fact thatthe variable , indicating whether full output feedbackor perturbation feedback is applied, does not appearin the linearized equations of motion [see (18)]. Moreover it isshown that the actuator forces, needed to stabilize the millingprocess, will be zero in steady state in case of perturbation feed-back whereas the actuator forces will be non-zero in steady statein case of full output feedback. Based on the previous discus-sion and the fact that for an AMB it is important to limit the inputcurrent in order not to exceed the maximum amount of carryingforce, in the remainder of this paper, perturbation feedback willbe considered. In practice, the perturbation displacementscan be obtained by using a chatter detection algorithm based ona parametric model of the milling process, as, e.g., described in[4].

V. MODELLING FOR ROBUST CONTROL DESIGN

The model of the milling process, discussed in Section II, canreadily be employed for stability analysis (i.e., determination ofthe SLD). However, the presence of time-delay and the explicittime-dependency of the right-hand side of the DDE (9) com-plicate the development of robust control synthesis techniques.Therefore, we apply two model simplifications to construct afinite-dimensional, time-invariant model, which will be moresuitable for controller synthesis. Moreover, the effect of thesemodel simplifications on the SLD is demonstrated.This section will discuss two model simplifications to con-

struct a finite-dimensional time-invariant model of the millingprocess, which will be more suitable for controller synthesis.


First, the discussion will focus on an autonomous approxima-tion of the linearized nonautonomous DDE describing the lin-earized perturbation milling dynamics, obtained by linearising(9) about , given by (12)

(19)

A characteristic feature of a milling process is that the directionof the cutting forces is a function of the rotation angle . As aresult, time-dependent functions appear in the describing modelequations. In [17], a method is described which approximates

by means of a Fourier series expansion. The number ofharmonics to be considered for an accurate reconstruction of

depends on the immersion conditions (which indicatesthe percentage of the tool diameter used during cutting) and thenumber of teeth in cut. In this paper, we will consider full im-mersion cuts (i.e., the entire tool diameter is used for cutting).Then, as described in [17], it is sufficient to take the average(zero-order) component of the Fourier series expansion over onetooth passing, i.e.

(20)

Since is valid only between the entry and exit anglesof the cutter (i.e., when ), it becomes equal to theaverage value of at cutter pitch angle

(21)

where the integrated functions , and can bedetermined analytically in case of a linear cutting model (, see [17]) and have to be computed numerically in case of anexponential cutting model .At this point we have obtained a time-invariant milling model

in which the dependency on the rotation angle is elimi-nated.Secondly, a finite-dimensional approximation of the time

delay, using a Pad approximation, is applied (see also [15]and [16] where Pad approximation is used for controllerdesign in case of turning). Hereto, the delayed tool vibrations

are approximated using a Padapproximation and the resulting approximation is denoted by

, such that . The choicefor a suitable order of the Pad approximation depends on the

Fig. 4. Stability lobes diagram for the milling process with Pad approximationof order and the milling model with exact time delay.

eigenfrequencies of the spindle-toolholder-tool dynamics. Inorder to accurately approximate the regenerative effect, given,in the frequency domain by , an approximationfunction should be chosen which has magnitude 1 at the fre-quencies , and accurately approximate the phaseof . Consequently, the order of the Pad approximationshould be chosen such that it approximates the phase ofup to at least the highest frequency of the spindle-toolholderand tool modes which are relevant for chatter.The milling model in (19) with cutting force averaging, de-

fined in (21) and Pad approximation is given as shown in (22) atthe bottom of the page, where

, and denote matrices of the state-space de-scription of the Pad approximation. The size of these matricesdepends on the chosen order for the Pad approximation.Since the delayed output vector has two elements( - and -direction), the state-space description of the Pad ap-proximation has two times the number of states of the Pad ap-proximation order . The order of the Pad approximation willbe based on a desired level of accuracy regarding the predictedchatter stability boundary using the model with Pad approxi-mation.In Fig. 4 the chatter stability boundary is given for the au-

tonomous model with time-delay and for different orders ofthe Pad approximant with the parameters of the model listedin Table I. From Fig. 4 it can be observed that, for increasingorder of the Pad approximant, the error between the sta-bility lobes determined using the exact delay term and the ap-proximated delay term becomes smaller. Moreover, since thedelay is inversely proportional to the spindle speed, the approx-imation becomes more accurate as the spindle speed increases.

(22)


Fig. 5. Generalized plant interconnection.

In this work we focus on high-spindle speeds (i.e., above 20krpm). Hence, it is sufficient to choose the order of the Pad ap-proximant for equal to be , which will be usedthroughout this paper.

VI. ROBUST CONTROLLER DESIGN

In the previous section, a model has been derived which issuitable for robust controller design. Therefore, in this sec-tion, the actual controller design for an active chatter controlmethodology, which will alter the chatter stability boundary, ispresented, such that stable operating points of higher produc-tivity can be attained while avoiding chatter. The control goal,as presented in Section III, can be cast into the generalized plantframework and solved using -synthesis techniques [26]. Fig. 5shows the configuration of this framework. The generalizedplant is a given system with three sets of inputs and three setsof outputs. The signal pair denote the inputs/outputs of theuncertainty channel. The signal represents an external input inwhich possible disturbances, measurement noise and referenceinputs are stacked. The signal is the control input. The outputcan be considered as a performance variable while denotes

the measured outputs used for feedback. The remainder of thissection will be devoted to uncertainty modelling, in depth of cutand spindle speed and the specification of the performance

requirement for the active chatter control problem as stated inSection III.

A. Nominal Model

Equation (22) gives the nominal plant model used during-synthesis. Note that, in contrast to most active chatter con-trol methods discussed in the introduction, in this work we donot only consider the spindle dynamics during the control de-sign, but also take the interaction between the spindle dynamicsand the cutting forces (and therewith the regenerative effect re-sponsible for chatter) into account. It is expected that this is amore profound and promising method to make dedicated modi-fications to the chatter stability boundary by means of feedbackcontrol.

B. Uncertainty Modelling

This section describes the modelling of the uncertainties inthe process parameters, which can be considered as a key stepin achieving the control objective defined above: robust stability

(i.e., chatter avoidance) in a predefined range of process param-eters. The control design will be based on the milling model (22)presented in the previous section.1) Uncertainty in Process ParameterDepth of Cut :

First, the uncertainty in depth of cut is considered which ismodelled as a parametric uncertainty. An important (practical)aspect is that robust control design should provide stability forsmall as well as (relatively) large values of the depth of cuts,see also [16]. Hereto, the uncertain depth of cut is modelledsuch that it specifies a range from zero up to a maximum value, i.e., . Let us define a real scalar uncertainty set

. The uncertainty for the depthof cut is then defined by

(23)

where is the maximal depth of cut for which stable cutting isdesired.2) Uncertainty in Process ParameterSpindle Speed :

Next, the uncertainty model for the spindle speed is considered.As described before, the delay is inversely proportional to thespindle speed. Hence, uncertainty in spindle speed is mod-elled as an uncertainty in the delay , where . Sincea Pad approximation is a rational function of two polynomialsin the Laplace operator and delay , modelling the intervaldelay via a parametric uncertainty would result in an overalluncertainty of very large dimensions (due to the relatively largeorder of the Pad approximations).Here an alternative approach will be used to model the delay

uncertainty. Hereto, note that for arbitrary frequency , thevalue set of the frequency-domain delay operator for all

can be represented in the complex plane as a circulararc extending along the unit circle. This time-delay intervalcan be approximated by choosing any pair of stable transferfunctions and such that , with

and , covers the uncertainty set with. Several alternatives exist to determine transfer

function and satisfying these conditions. Chenand Knospe [16] propose to choose and suchthat at each frequency: 1) the arc length covered by the disk

is nearly that of the delay element , forand 2) the area of the disk lying outside the unit circle

is minimized. Doing so results in a transfer functionwhich has twice the order of the chosen Pad approximation.Since the Pad approximation needed to accurately describe thedelay term is already of a relatively high order, the generalizedplant will be of an even higher order which is not desired dueto possible computational and implementation issues for theresulting controller. Moreover, the size of the circle coveringthe circular arc of the delay uncertainty is rather large which isdue to the fact that the area of the disk lying outside the unitcircle is chosen to be minimized. This approach may thereforegive conservative results as illustrated for the milling processin [1].Hence, here a different approach is presented to model the

delay uncertainty. In contrast to the approach discussed above,


we model the delay uncertainty based on a Pad approxima-tion of the nominal model. The total delay uncertainty intervalis then overapproximated using a low-order transfer functionwhich covers the circular arc of the delay uncertainty intervalalong the unit disk about the nominal delay. Hereto, consider thelinearized autonomous milling model with a delay uncertaintyonly. Basically, this model can be represented by the followingstate-space model:

(24)

where , ,and uncertainty set . It is easy to show that (24) canbe written as a feedback interconnection between the dynamics

(25)

and uncertainty term

(26)

where , the delay operator is defined asand . The representation of

the time-domain operator in (26) can be given in the Laplacedomain as

(27)

where , the Laplace transforms of and ,respectively. Let be the gain bound of the uncertainty op-erator (27) in the frequency domain, given as

(28)

Since the transfer function is analytic andbounded in the open right half of the complex plane, the

-norm of can be determined by evaluatingthe transfer function on the imaginary axis, i.e., for .Consequently, in order to determine a bound on ,it should be determined for . It can be shown, see [27],that the upper bound on the delay uncertainty is repre-sented as follows:

(29)

where . The frequency-dependent upper boundon the delay uncertainty is not a rational function and

can therefore not readily be used during controller synthesis.

Hereto, in [27] rational transfer functions for several or-ders are derived such that . Since,in this work, the high-speed milling process is considered, thedelay intervals will be relatively small (typically of )s. for typical spindle speed ranges of rpm for spindlespeeds rpm). Using this fact, together with the factthat the dominant spindle dynamics resonances lie in generalbetween Hz, implies that typi-cally and, consequently, an accurate approximationof the frequency-dependent upper boundis required. Moreover, from a numerical point of view, propertransfer functions are desired. Then, based on the resultsin [28], is chosen as

(30)

which ensures a tight over bound of (by ) espe-cially in the frequency region which is relevant in the case ofhigh-speed milling, as is described above. Hence, by using theresults presented above, the delay uncertainty is approximatedby two rational transfer functions and , where

is the Pad approximation of and ,with as in (30), such that

(31)

C. Performance Requirement

This section discusses the specification of a performance re-quirement for the active chatter control design. In essence, thechatter control problem at hand is a robust stabilization problemrather than a performance problem. As outlined in the problemstatement in Section III, the robust stability requirement has tobe achieved with limited control effort, since actuator forceshave to satisfy practical saturation limits (of, e.g., AMB). There-fore, the control gain should be bounded during -synthesis,which reflects the most relevant performance requirement forchatter control.Limiting the control gain is done by applying an

upper bound on the control sensitivity transfer function, where we have the

equation at the bottom of the page, which gives the transferfunction representation from to of the nominal plantgiven by (22). Here, the control sensitivity is defined as thetransfer function from a input signal (which can, e.g., beinterpreted as measurement noise on the measured perturba-tion displacements entering the feedback loop) to thecontrol input . The bound on the control sensitivity is


Fig. 6. Block diagram of linearized approximated autonomous milling modelwith performance weighting.

enforced by defining a weighting function , whichwill be described below, such that the performance outputof the generalized plant is the weighted control sensitivity

. A schematic overview of the closed-loopapproximated autonomous milling model with performanceweighting is given in Fig. 6 where the transfer function of thespindle-actuator dynamics is denoted by and given by

(32)

Then, the problem in which it is aimed to find a -optimal con-troller which stabilizes the milling process in the face ofmodelled uncertainties (in and ) while minimizing the peakmagnitude of the weighted control sensitivity; that is a con-troller which achieves .Of course, minimizing the weighted control sensitivity actuallyenforces a frequency-dependent upper bound on the magnitudeof control gain rather than on the magnitudes of the ac-tual control input . Hence, by estimating the magnitude ofthe inputs to the controller, i.e., of the chatter-related tool dis-placements in - and -direction, an appropriate bound on thecontrol gain can in practice be chosen such that the actuatorforces are satisfying given saturation limits. In this work, theweighting transfer function matrix is chosen to be di-agonal, because of the 2-D nature of the control input , i.e.,

. Moreover, its structureis chosen such that is a lead-lag filter with high-passcharacteristics. This means that, for frequencies below theroll-off frequency , the control gain is limited by acertain value and that, for frequencies larger than , the in-puts to the controller are attenuated in order to reduce (unde-sired) influences on the control action, due to sensor noise andaliasing effects due to discretization of the continuous-time con-troller. These (undesired) influences need to be attenuated forany practical (digital) implementation of the controller. More-over, in general, an actuator has a limited bandwidth in which itcan operate. Consequently, the roll-off frequency should bechosen such that it accounts for the actuator bandwidth limita-tion, high-frequentmeasurement noise sensor noise and respectssampling frequency influences. Based on the discussion above,weighting function is written as

(33)

where denotes the gain of the weighting function. A pole,at frequency (such that ), is added to obtaina proper weighting function, necessary for implementation.

As often in robust control, is chosen in an iterativefashion.

D. Generalized Plant FormulationBased on the discussion on uncertainty modelling and the

specification of a performance requirement in the previous sec-tions and the milling model for control, presented in Section V,the control problem can now be transformed into the general-ized plant framework [26].In order to derive the generalized plant, first, consider the

following state-space descriptions of the systems , ,where and , respectively

(34)

(35)

(36)

where the size of depends on the order of the Pad approx-imation which should be chosen such that it approximates thephase of up to at least the highest resonance frequencyof the spindle-toolholder and tool modes which are relevant forchatter. Next, consider the linearized autonomousmillingmodelas described by (22)

(37)

Then, by adding the uncertainty and performance channelinput/output, denoted by and , respectively,to the system and rearranging terms, the generalized plant isgiven as follows:

(38)

with the state vector ,input vector , output vector

. The uncertainty channel inputand output are defined as

where the subscripts and denote the input/output of thedelay and depth of cut uncertainty, respectively. The definitionof the state-space matrices of the generalized plant can be foundin the appendix of the paper. Combining all the sources of uncer-tainty as described in Section VI-B, the total uncertainty blockis given as

(39)

From the definition of the generalized plant and corre-sponding uncertainty set it becomes clear that the control


problem at hand is a robust performance problem which con-tains structured uncertainties, i.e., the uncertainty is not a fullcomplex matrix but has specific elements which contain uncer-tainties. Hence, it is recommended to solve the problem using-synthesis, which will be briefly discussed in the followingsection.

E. Controller SynthesisIn this section, the problem of finding controllers, which sat-

isfy the requirements as defined in Section III, will be discussed.As discussed in the previous section, the control problem at handis a robust performance problem. It is well known that thereis no direct method available yet to synthesis a -optimal con-troller, see [26]. From the generalized plant model, presentedin the previous section, it can be concluded that we are dealingwith a so-called mixed -synthesis problem, i.e., both complexand real uncertainties are present. Although mixed -synthesiscan be employed via D,G-K-iteration, it will in general resultin high-order controllers due to high-order fits required for theG-scales. Moreover, as demonstrated in [13], D,G-K-iterationdoes not guarantee an increase in performance. As the generalplant in this work is of relatively high order (since a relativelyhigh-order Pad approximation is needed to accurately approx-imate the time delay), the uncertainty in depth-of-cut is consid-ered as a complex uncertainty and controller design is employedusing D-K-iteration. We accept the possible conservatism in-troduced by considering only complex uncertainties during thecontroller design in order to avoid the design of a controller ofeven higher order. The complex uncertainty set, denoted by ,is given as

(40)

By using an upper bound on the structured singular value ,the controller synthesis problem is transformed into an opti-mization problem which tries to minimize the peak value overfrequency for this upper bound, namely

(41)

Herein, denotes a (frequency-dependent) scaling matrix torepresent an upper bound on and denotes the setof functions that are analytic and bounded in the open right halfplane. The optimization problem (41) is iteratively solved forand . For a fixed scaling transfer matrix , the problem re-

duces to a standard synthesis problem, which can be turnedinto a convex optimization problem. The optimization problemfor a fixed controller matrix, i.e., the problem of determiningthe optimal scaling matrix for a given frequency can alsobe recast in to a convex optimization problem. Both the aswell as the step in the D-K-iteration can be solved using al-gorithms from the Robust Control Toolbox of MATLAB [29].

F. Controller Order ReductionDue to the relatively high order of the Pad approximation

needed to accurately describe the delay term, the resulting con-trollers will be of relatively high order , see [1]. Asdiscussed above, the spindle dynamics typically has resonances

TABLE IIPARAMETERS OF THE AMB MODEL

lying between Hz, which will gener-ally result in relatively fast controller poles which in turn requirerelatively large sample frequencies in a digital implementation.Hence, for the purpose of the feasibility of the implementationof the proposed active chatter control methodology in practice,controller order reduction should be applied.Balanced truncation is an order reduction procedure which is

often applied to tackle such model reduction problems. How-ever, balanced truncation can only be applied in case the systemto be reduced is stable. The control synthesis procedure dis-cussed in the previous section does, however, not guarantee thedesign of stable controllers. To deal with this fact, closed-loopbalanced truncation can be applied, see [30].The controller states which do not contribute significantly to

the closed-loop input/output of the generalized plant will be re-moved from the controller using closed-loop balanced trunca-tion. After that, robust performance for the closed-loop systemwith the reduced-order controller is evaluated by determining

-values. The acceptable amount of reduction is defined asthe smallest controller order for which .As already outlined above, the robust control problem under

consideration has structured uncertainties, which will be solvedvia D-K-iteration. Hence, during closed-loop balanced trunca-tion, the D-scaling matrices obtained during controller synthesisare absorbed into the generalized plant.

VII. RESULTSIn this section, the results for actual controller synthesis for a

realistic model of a high-speed milling machine is addressed. Inorder to demonstrate the feasibility of the -synthesis approachproposed in the previous section, control design is performedfor an illustrative example. Hereto, consider the parametersof the milling process as given in Table I. The spindle dy-namics is modelled, as before, by two decoupled subsystemsconsisting of a two mass-spring-damper model in order tocapture the inherent compliance between the actuator/sensorsystem (with mass ) and the cutting tool (withmass ). The parameters of the AMB model arelisted in Table II. Moreover, a four-fluted tool is considered.Consequently, as already discussed in Section V, a 10-th orderPad approximation is used to approximate the time delay inthe milling model.We aim to design controllers that stabilize milling operations

(i.e., guarantee the avoidance of chatter) for two differentspindle speed ranges, for a range of depth-of-cut whichshould be as large as possible for a given performance require-ment (i.e., for a given limitation of the control gain). Hereto,D-K-iteration is employed within a bi-section scheme.The performance requirement, presented in the previous sec-

tion, is used to limit the control forces. For an AMB it is im-portant to limit the input current in order not to exceed themaximum amount of carrying force. Here, we choose to limit


Fig. 7. Closed-loop -values for reduced controllers using closed-loop balanced truncation. Black bars indicate an unstable closed-loop: (a)rpm; (b) rpm.

the input current to 2 A. Typical bearing displace-ments , for the modelled spindle under consideration, re-lated to onset of chatter, are of the order of mm. Then,an upper bound on the control gain due to physical constraintsof the actuator is set to N/mm and con-sequently in (33) is set equal to mm/A. Thehigh-frequent roll-off frequency is chosen to account foractuator bandwidth limitation (which is typically a few kHzin case of an AMB), high-frequent sensor noise and aliasingeffects due to discretisation of the continuous-time controller(sampling frequencies will be typically set to a value larger than15 kHz). Based on this is set to 7500 Hz. Theremaining parameter of the weighting filter is set to

Hz, the additional pole is added such that, first,is well-posed and, second, the generalized plant fulfills

the rank conditions typically made in the problem, see [26,p. 354].Controllers are designed for two different ranges

of spindle speeds, namely a relatively small interval givenas rpm and a relatively large intervalgiven as rpm. Controller synthesis usingD-K-iteration yields a 40-th order controller for a maximaldepth of cut of mm for

rpm and a 36-th order controller for a max-imal depth of cut of mm for

rpm. The difference between the controllerorders is due to a difference in the D-scales.In Fig. 7, the -value for different controller orders (after

reduction) are depicted for the two ranges of spindle speeds.In general, closed-loop stability cannot be guaranteed aftercontroller order reduction. Therefore, before determining the

-value for a specific reduced-order controller, first stabilityof the nominal closed-loop system is checked. When theclosed-loop system is stable the corresponding -value isdetermined. From these results, the lowest controller order isselected for which robust performance can be guaranteed, i.e.,the lowest controller order for which . Thisyields a 24-th order controller for anda 16-th order controller for rpm. So, itcan be concluded that a significant reduction of the controllerorder can be achieved while still guaranteeing robust stability

and performance which warrant the feasibility of the practicalimplementation of such controllers in practice.Frequency response functions (FRF) of the full- and re-

duced-order controllers together with the inverse of thefrequency bound imposed on the control sensitivity (i.e.,

) are given in Fig. 8. It can be seen that the re-sulting controllers exhibit highly dynamical characteristicsindicated by the inverse notches in the FRF. Although themagnitude of the controllers do not exactly fulfill the imposedbound (since the bound is imposed on the control sensitivity

and not the controlleritself), it can be seen that the magnitude is bounded.

Moreover, it can be seen that the full- and reduced-ordercontroller have similar FRF magnitudes. Hence, it is expectedthat robust performance is maintained under controller-orderreduction. To verify whether robust performance is maintained,stability lobes diagrams (SLDs) are determined using the(closed-loop) linearized non-autonomous milling model (18),as outlined in Section II-D, for the case with the reduced-ordercontroller and without control. Note that for the case withoutcontrol, (12) and (13) are used for the stability analysis. The re-sulting SLDs can be found in Fig. 9. It can be seen that the SLDof the controlled milling are shaped such that it contains a lobein the desired spindle speed range. Stability is ensured up to adepth of cut 3.055 mm (an increase of approximately760% compared to the case without control) and2.686 mm (an increase of approximately 241% compared tothe case without control) where controllers are designed for

rpm and rpm, respec-tively. Herein, denotes the maximal achievable depth ofcut in the SLD in the desired spindle speed range. Fig. 9 clearlyillustrates the power of the proposed approach, as the SLD isshaped locally to be able to increase at a specific spindlespeed (while avoiding chatter and satisfying a specified boundon the control gain). This is contrary to the application of activedamping which lifts the SLD over the entire spindle speedrange at the cost of high required levels of actuation energy.Whereas stability is increased at the desired spindle speeds, itdecreases significantly at other spindle speeds.The characteristics of the controller design and its ability to

shape the SLD in a dedicated fashion can be explained by fur-


Fig. 8. Magnitude of FRF of the full-order (black) and reduced-order (grey) controllers obtained by D-K-iteration for two different range of spindle speedsand rpm.

Fig. 9. Stability lobes diagrams, determined using the linearized nonau-tonomous milling model (18), for reduced-order controllers designedfor two different range of spindle speeds, and

rpm.

ther examining the controlled spindle dynamics. The FRF of theclosed-loop tool-tip spindle dynamics (i.e., the FRFfrom to ) is given, together with the original (uncon-trolled) spindle dynamics, in Fig. 10. While the original (un-controlled) spindle dynamics only has - and -components(due to decoupled spindle dynamics), the controlled machinedynamics also has off-diagonal components. This can be ex-plained by the fact that controller design is performed using thecomplete milling model where coupling between - and -di-rection is introduced by the cutting force model (resulting in afull matrix in (21) and consequently in a full 2 2 controller

). A striking characteristic displayed in Fig. 10 is the factthat the controller has tailored the spindle dynamics such thatthe resonances are shifted. A similar conclusion was drawn in[16] for active chatter control in case of the turning process.For the small spindle speed range rpm ,a dominant weakly damped resonance can be seen which is lo-

cated at 1667 Hz. A better damped resonance around2000 Hz is created in case of the larger spindle speed range

rpm , which lies near the edge of the range of de-sired tooth passing frequencies. As a matter of fact, the locationof the closed-loop resonances, which are dominant for chatteroccurrence, is closely related to the tooth excitation frequencies

for milling operations within the defined spindlespeed ranges. Hence, it can be concluded that, in order to createa stability lobe at a certain spindle speed, the natural frequencyof the spindle dynamics should be set equal the correspondingtooth passing excitation frequency, see also [18]. The fact thata closed-loop spindle resonance situated at a tooth-passing ex-citation frequency is beneficial for avoiding chatter can be ex-plained as follows. In the milling process the highest depth ofcut can be obtained (corresponding to a peak in the SLD) whenthe dynamic chip thickness is equalto zero. This relation can be transformed to the frequency do-main as follows:

(42)

where and are the Fourier transforms ofand , respectively. Hence, the difference between

the tooltip displacements of the present and previous cut isactually characterized by a filter, denoted by , with zerosat . Moreover, for the millingprocess, the dominant (chatter) frequency of the perturbationvibrations lies in general close to the eigenfrequency of thespindle dynamics. Then, by designing the controller such thatthe closed-loop resonance is close to a tooth-passing frequencyand due to the filter properties of the (in particular thelocation of the zeros of at -related frequencies), thedynamic chip thickness is enforced to be zero at the desiredspindle speed. This, in turn, results in a large depth of cutwithin the desired spindle speed range and a peak in the SLDat that spindle speed. So, by applying robust control designtechniques, a controller is obtained which tailors the tooltipspindle dynamics, such that a resonance is created near a toothpassing harmonic which in turn results in a peak in the SLD.


Fig. 10. Controlled (black) and uncontrolled (open-loop) (grey) tooltip spindle dynamics for reduced-order controllers designed for the twodifferent range of spindle speeds, and rpm. The interval of tooth passing excitation frequencies corresponding to thespindle speed range is indicated by the grey area.

Fig. 11. Displacements at the tooltip for a time domain simulation performed for 31 000 rpm and 2 mm with and without control using thenonlinear AMB model. The controller is designed for a spindle speed range of rpm. (a) Control off. (b) Control on, .

In the final part of this section some results from time-do-main simulations (TDS) will be discussed. Hereto, the non-linear nonautonomous delay differential equations describingthe total milling model, given by (9), along with the non-linear bearing model, given by (5), have been implemented inMATLAB/SIMULINK. The purpose of the TDS is to furtherillustrate the effectiveness of the controller design, the benefitof perturbation feedback and investigate whether it is justifiedto apply the linear AMB model for controller design.As described before, for an AMB it is important to limit the

controller input in order to avoid actuator saturation. In orderto apply perturbation feedback, the periodic solution withperiod time has to be known. Existence of the periodic solutionin case of the nonlinear bearing model is difficult to prove. Here,the periodic solution is approximated using the finite differencemethod, as illustrated for the milling model in [23]. In practice,the perturbation displacements can be obtained by using achatter detection algorithm based on a parametric model of themilling process, as, e.g., described in [4]. For the sake of brevity,we do not describe these algorithms here.In order to compare the performance of the milling process

with and without chatter control the simulation is performedfor an operating point which is originally unstable (i.e., exhibitschatter), but is stabilized by means of control. Here, the opera-

tion point under consideration has the process parameters31 000 rpm and 2 mm, which is originally an unstableworking point (see Fig. 9). The results are gathered in Figs. 11and 12. Fig. 11 presents the displacements at the tooltip withcontrol off and control on with perturbation feedback .Furthermore, the -sampled tool displacements is depicted bydots. Fig. 12 presents the actuator currents generated by the con-troller for the case with control on. From the figures, it can beseen that without control, the amplitude of the displacementsbecomes relatively large (approximately 35% of the tool radiuswhich is chosen as 5 mm), which will result in the tool jumpingin and out of cut resulting in an inferior workpiece quality.Whenthe controller is switched on, the motion for the initially un-stable working point is stabilized, which can be seen from the-sampled displacements. Moreover, it can be seen that the am-plitude of the displacements is considerably smaller for the casewith active chatter control as compared to the uncontrolled case.The actuator input currents are given in Fig. 12. Due to the factthat perturbation feedback is applied, the (steady-state) actuatorcurrent, after some transients at the start of the simulation, are(almost) zero. From the results of the time-domain simulations,it can be seen that the assumptions, for which the linear AMBmodel is a good approximation of the nonlinear AMBmodel, asdiscussed in Section II-B, remain valid. Moreover, the results


Fig. 12. Actuator input currents for a time domain simulation per-formed for 31 000 rpm and 2 mm where perturbation feedbackis considered. The controller is designed for a spindle speed range of

rpm.

from time-domain simulations clearly demonstrate the benefitof applying perturbation feedback.

VIII. CONCLUSION

In this paper, an active chatter control designmethodology forthe suppression of regenerative chatter in the high-speed millingprocess has been developed. The main purpose achieved is thesuppression of chatter (i.e., stabilization of the milling process)in an a priori specified range of process parameters (spindlespeed and depth of cut), such that working points of signifi-cantly higher productivity become feasible while avoiding un-desirable chatter vibrations. Herein, the requirement for a prioristability guarantee for a predefined range of process parametersis cast into a robust stability requirement. Moreover, a perfor-mance requirement is imposed on the control sensitivity in orderto limit the actuator forces. Current chatter control strategies forthe milling process cannot provide such a strong guarantee of apriori stability for a predefined range of working points. Thecontrol problem is solved via -synthesis using D-K-iteration.

In addition, it is shown that the actuator forces, needed to sta-bilize the milling process, will be zero in steady state in case ofperturbation feedback (i.e., only chatter vibrations are used as afeedback signal) whereas the actuator forces will be non-zeroin steady state in case of full output feedback. This result isexploited for a milling model incorporating a nonlinear ActiveMagnetic Bearing model, where it is important to limit the actu-ator input current in order to avoid actuator saturation. Results,for illustrative examples, clearly illustrate the power of the pro-posed control methodology. The chatter stability boundary islocally shaped to stabilize the desired range of working points.This is contrary to the application of active damping which liftsthe SLD over the entire spindle speed range at the cost of highrequired levels of actuation energy. By means of illustrative ex-amples it is shown that this control strategy can render workingpoints of significantly higher productivity stable.

APPENDIXSpindle Rotor Dynamics: The state-space matrices of the

spindle rotor dynamic model, given in (4), are defined as fol-lows:

where

for

Generalized Plant: The state-space matrices of the gener-alized plant, given in (38), are defined as shown in the equationat the bottom of the page.


REFERENCES

[1] N. J. M. van Dijk, N. van de Wouw, E. J. J. Doppenberg, J. A. J. Oost-erling, and H. Nijmeijer, Chatter control in the high-speed millingprocess using -synthesis, in Proc. Amer. Control Conf., 2010, pp.61216126.

[2] S. Smith and T. Delio, Sensor-based chatter detection and avoidanceby spindle speed, J. Dyn. Syst., Meas. Control, vol. 114, no. 3, pp.486492, 1992.

[3] R. E. Haber, C. R. Peres, A. Alique, S. Ros, C. Gonzalez, and J. R.Alique, Toward intelligent machining: Hierarchical fuzzy control forthe end milling process, IEEE Trans. Control Syst. Technol., vol. 6,no. 2, pp. 188199, Mar. 1998.

[4] N. J. M. van Dijk, E. J. J. Doppenberg, R. P. H. Faassen, N. van deWouw, J. A. J. Oosterling, and H. Nijmeijer, Automatic in-processchatter avoidance in the high-speed milling process, J. Dyn. Syst.,Meas. Control, vol. 132, no. 3, p. 031006(14), 2010.

[5] A. Yilmaz, E. AL-Regib, and J. Ni, Machine tool chatter suppressionby multi-level random spindle speed variation, J. Manuf. Sci. Eng.,vol. 124, no. 2, pp. 208216, 2002.

[6] E. Soliman and F. Ismail, Chatter suppression by adaptive speed mod-ulation, Int. J. Mach. Tools Manuf., vol. 37, no. 3, pp. 355369, 1997.

[7] T. Insperger and G. Stpn, Stability analysis of turning with peri-odic spindle speed modulation via semi-discretisation, J. Vibr. Con-trol, vol. 10, pp. 18351855, 2004.

[8] K. J. Liu and K. E. Rouch, Optimal passive vibration control of cuttingprocess stability in milling, J. Mater. Process. Technol., vol. 28, no.12, pp. 285294, 1991.

[9] Y. S. Tarng, J. Y. Kao, and E. C. Lee, Chatter suppression inturning operations with a tuned vibration absorber, J. Mater. Process.Technol., vol. 105, no. 1, pp. 5560, 2000.

[10] J. L. Dohner, J. P. Lauffer, T. D. Hinnerichs, N. Shankar, M. E. Regel-brugge, C. M. Kwan, R. Xu, B. Winterbauer, and K. Bridger, Mitiga-tion of chatter instabilities in milling by active structural control, J.Sound Vibr., vol. 269, no. 12, pp. 197211, 2004.

[11] S. Kern, C. Ehmann, R. Nordmann,M. Roth, A. Schiffler, and E. Abele,Active damping of chatter vibrations with an active magnetic bearingin a motor spindle using -synthesis and an adaptive filter, presentedat the 8th Int. Conf. Motion Vibr. Control, Daejeon, Korea, 2006.

[12] S. Kern, A. Schiffler, R. Nordmann, and E. Abele, Modelling and ac-tive damping of a motor spindle with speed-dependent dynamics, pre-sented at the 9th Int. Conf. Vibr. Rotat. Mach., Exeter, Great Britain,2008.

[13] R. L. Fittro, C. R. Knospe, and L. S. Stephens, -synthesis appliedto the compliance minimization of an active magnetic bearing HSMspindles thrust axis, Machin. Sci. Technol., vol. 7, no. 1, pp. 1951,2003.

[14] Y. Zhang and N. D. Sims, Milling workpiece chatter avoidance usingpiezoelectric active damping: A feasibility study, Smart Mater. Struc-tures, vol. 14, no. 6, pp. N65N70, 2005.

[15] M. Shiraishi, K. Yamanaka, and H. Fujita, Optimal control of chatterin turning, Int. J. Mach. Tools Manuf., vol. 31, no. 1, pp. 3143, 1991.

[16] M. Chen and C. R. Knospe, Control approaches to the suppression ofmachining chatter using active magnetic bearings, IEEE Trans. Con-trol Syst. Technol., vol. 15, no. 2, pp. 220232, Mar. 2007.

[17] Y. Altintas, Manufacturing Automation. Cambridge, U.K.: Cam-bridge Univ. Press, 2000.

[18] Y. Altintas and M. Weck, Chatter stability of metal cutting andgrinding, CIRP AnnalsManuf. Technol., vol. 53, no. 2, pp.619642, 2004.

[19] R. Faassen, Chatter prediction and control for high-speed milling:Modelling and experiments, Ph.D. dissertation, Dept. Mech. Eng.,Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2007.

[20] G. Stpn, Modelling nonlinear regenerative effects in metal cutting,Philosophical Trans. Royal Soc. A: Math., Phys., Eng. Sci., vol. 359,no. 1781, pp. 739757, 2001.

[21] R. P. H. Faassen, N. van de Wouw, J. A. J. Oosterling, and H. Ni-jmeijer, Prediction of regenerative chatter by modelling and analysisof high-speed milling, Int. J. Mach. Tools Manuf., vol. 43, no. 14, pp.14371446, 2003.

[22] G. Schweitzer, H. Bleuler, and A. Traxler, Active Magnetic Bear-ings. Zrich: Verlag der Fachvereine, 1994.

[23] R. P. H. Faassen, N. van de Wouw, H. Nijmeijer, and J. A. J. Ooster-ling, An improved tool path model including periodic delay for chatterprediction in milling, J. Comput. Nonlinear Dyn., vol. 2, no. 2, pp.167179, 2007.

[24] M. Farkas, Periodic Motions. Berlin: Springer-Verlag, 1994.[25] T. Insperger and G. Stpn, Updated semi-discretization method for

periodic delay-differential equations with discrete delay, Int. J. forNumer. Methods Eng., vol. 61, no. 1, pp. 117141, 2004.

[26] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control:Analysis and Design, 2nd ed. Chichester, U.K.: Wiley, 2005.

[27] Y.-P. Huang and K. Zhou, Robust stability of uncertain time-delaysystems, IEEE Trans. Autom. Control, vol. 45, no. 11, pp. 21692173,Nov. 2000.

[28] Z.-Q. Wang, P. Lundstrm, and S. Skogestad, Representation of un-certain time delays in the H1 framework, Int. J. Control, vol. 59, no.3, pp. 627638, 1994.

[29] The MathWorks, Inc., Natick, MA, Documentation, 2010. [Online].Available: http://www.mathworks.com

[30] C. Ceton, P. M. R.Wortelboer, and O. H. Bosgra, Frequency weightedclosed-loop balanced reduction, in Proc. 2nd Euro. Control Conf.,1993, pp. 697701.

Niels van Dijk received the M.Sc. and Ph.D. degreesin mechanical engineering from the Eindhoven Uni-versity of Technology, Eindhoven, The Netherlands,in 2006 and 2011, respectively.Currently he is with the Mechatronics Technolo-

gies Group, Philips Innovation Services, Eindhoven,The Netherlands. His interests include modelling andcontrol of manufacturing and high-precision systems.

Nathan van de Wouw received the M.Sc. (withhonors) and Ph.D. degrees in mechanical en-gineering from the Eindhoven University ofTechnology, Eindhoven, The Netherlands, in 1994and 1999, respectively.From 1999 to 2010, he has been an Associate

Professor with the Department of Mechanical En-gineering, Eindhoven University of Technology. Hewas with Philips Applied Technologies, Eindhoven,The Netherlands, in 2000 and at the NetherlandsOrganization for Applied Scientific Research, The

Netherlands, in 2001. He was a visiting Professor with the University ofCalifornia Santa Barbara, in 2006/2007 and the University of Melbourne,Australia, in 2009/2010. He has published a large number of journal andconference papers and the books Uniform Output Regulation of NonlinearSystems: A convergent Dynamics Approach with A.V. Pavlov and H. Nijmeijer(Birkhauser, 2005) and Stability and Convergence of Mechanical Systems withUnilateral Constraints with R.I. Leine (Springer-Verlag, 2008).

Ed J. J. Doppenberg received the B.S. degree incontrol engineering from the Rotterdam University,the Netherlands, in 1974 and the M.Sc. Ing-degreein control engineering from the Delft University ofTechnology, Delft, the Netherlands, in 1989.He has been working from 1987 until 2002

in the development of active noise and vibrationadaptive control system for the transportation sector(aerospace and automotive applications) at TNOScience and Industry, Delft, the Netherlands. He wasinvolved from 2004 to 2010 with the manufacturing

technology development at TNO Science and Technology. In 2011 he joinedthe Department of Precision Motion Systems, TNO. He has published severalpapers on adaptive (MIMO) control systems and on machining technology,chatter control and micro milling.


Han Oosterling received the M.Sc. degree in me-chanical engineering from the Eindhoven Universityof Technology, Eindhoven, TheNetherlands, in 1981.From 1981 to 1992, he has been with Manu-

facturing Technology Development, Dutch Navy,Philips Tool and Die Shop, and Philips Research. In1991, he started with TNO Science and Technology,Delft, The Netherlands, as a Research Leader of theMachining Technology Research Group. In 2011, hejoined the Department Space and Science, TNO. Hehas published several papers on coating technology,

machining technology, chatter control, and micro milling.

Henk Nijmeijer (F99) received the M.Sc. andPh.D. degrees in mathematics from the Universityof Groningen, Groningen, The Netherlands, in 1979and 1983, respectively.From 1983 to 2000, he was with the Department

of Applied Mathematics, University of Twente,Enschede, The Netherlands. Since 1997, he wasalso part-time affiliated with the Department ofMechanical Engineering, Eindhoven University ofTechnology, Eindhoven, The Netherlands, wheresince 2000, he has been a Full Professor with and

chairs the Dynamics and Control section. He has published a large numberof journal and conference papers and several books, including the clas-sical Nonlinear Dynamical Control Systems (Springer, 1990) coauthoredwith A. J. van der Schaft, Synchronization of Mechanical Systems (WorldScientific, 2003) coauthored with A. Rodriguez, Dynamics and Bifurcationsof Non-Smooth Mechanical Systems (Springer-Verlag, 2004) coauthored withR. I. Leine, and Uniform Output Regulation of Nonlinear Systems (Birkhauser2005) coauthored with A.Pavlov and N. van de Wouw.Dr. Nijmeijer is Editor-In-Chief of the Journal of Applied Mathematics, cor-

responding editor of the SIAM Journal on Control and Optimization, and boardmember of the International Journal of Control, Automatica, Journal of Dy-namical Control Systems, International Journal of Bifurcation and Chaos, In-ternational Journal of Robust and Nonlinear Control, Journal of Nonlinear Dy-namics, and the Journal of Applied Mathematics and Computer Science. He wasa recipient of the IEE Heaviside premium in 1990. In the 2008 research eval-uation of the Dutch Mechanical Engineering Departments the Dynamics andControl Group was evaluated as excellent regarding all aspects (quality, pro-ductivity, relevance, and viability).

milling process

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