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CIRP Annals - Manufacturing Technology 65 (2016) 373–376

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A trajectory optimization method for improved tracking of motioncommands using CNC machines that experience unwanted vibration

C. Okwudire *, K. Ramani, M. Duan

Mechatronics and Sustainability Research Laboratory, University of Michigan, Ann Arbor, USA

Submitted by T. Hoshi (1), Toyohashi, Japan

1. Introduction

Computer numerical control (CNC) powered feed drives areused for generating motion in a wide range of manufacturingmachines, e.g., machine tools, laser cutters and 3D printers.Therefore, their speed and accuracy are critical to the quality andproductivity of the associated manufacturing machines andprocesses [1]. Tracking and contouring performance are twometrics commonly used to quantify the accuracy of CNC feeddrives [2]. Tracking accuracy refers to how well the actual positionof a feed drive follows a desired trajectory, while contouringaccuracy is a measure of how close the actual position keeps to adesired motion path [2]. Most CNC feed drives achieve goodtracking accuracy (directly) and contouring accuracy (indirectly)by minimizing axis tracking errors [2]. Therefore, this paperfocuses on improving axis tracking accuracy, assuming thatcontouring accuracy is also indirectly enhanced.

It is not uncommon for a CNC-driven machine to have one ormore unwanted vibration modes, arising from structural flexibil-ities of its mechanical components. For example, vibration modescould stem from ball screws, machine columns/beds, or even fromfixtures added to a machine. Such unwanted vibration modes areoften excited by motion commands sent to the CNC machine,resulting in degradation of tracking accuracy.

One approach to reduce the loss of tracking accuracy caused bymotion commands is to suppress the resulting vibration of themachine using feedback controllers [3,4]. However, the presence of

An alternative approach, that does not require extra hardwnor incur stability issues, is to generate the motion commands sto the CNC such that unwanted vibration is avoided or reduced.most popular approach for achieving this goal is to genemotion commands that are smooth (i.e., commands that have lor no high-frequency content) [1]. However, the attenuated hfrequency content of smooth motion commands often implies

of motion speed, which adversely affects productivity [1]. Anotproblem with smooth motion commands is that, because theynot generated using knowledge of the machine’s dynamics, toften yield suboptimal vibration reduction [8]. Better vibrareduction can be achieved by using filters (e.g., FIR filters) or inshapers [8–10] to generate motion commands with low frequecontent around the resonance modes of the machine (assumthat its dynamics is known). However, filters and input shapintroduce time delays which sacrifice productivity and coseverely degrade tracking and contouring accuracy [8–10].

Altintas and Khoshdarregi [9] demonstrated a method

reducing input-shaper-induced contour errors by predicting

compensating for the errors via modifications to the mocommands sent to the machine. However, their method canguarantee the minimization of errors, nor can it ensure

kinematic constraints imposed on the motion trajectory are

violated. Moreover, it does not eliminate the loss of motion spcaused by the delays of the input shaper.

This paper presents a trajectory optimization method

minimizes tracking errors in CNC machines that have unwan

A R T I C L E I N F O

Keywords:

Computer numerical control (CNC)

Vibration

Trajectory optimization

A B S T R A C T

This paper presents a trajectory optimization method for minimizing tracking errors in CNC mach

that experience unwanted vibration. The motion command sent to the machine is parameterized u

B-splines whose basis functions are filtered using a model of the machine’s dynamics. The control po

associated with the filtered basis functions are then selected to minimize tracking errors subject to

kinematic limits of the machine. The proposed method is shown to be very systematic and easy to

and it does not sacrifice tracking speed. Experiments are used to demonstrate its superior performa

compared to an existing method.

� 2016 C

Contents lists available at ScienceDirect

CIRP Annals - Manufacturing Technology

journal homepage: http: / /ees.elsevier.com/cirp/default .asp

theonstrolteds of

vibration modes often introduces stability concerns in feedbackloops, especially when sensors and actuators are non-collocated[3]. Another approach is to use vibration suppression devices [5–7],requiring that the machine be furnished with extra hardware,which may not be practical nor desirable.

aticn 2

* Corresponding author.

E-mail address: okwudire@umich.edu (C. Okwudire).

http://dx.doi.org/10.1016/j.cirp.2016.04.100

0007-8506/� 2016 CIRP.

vibration modes. The interpolated motion commands sent tomachine are parameterized using B-splines whose basis functiare filtered using a model of the machine’s dynamics. The conpoints associated with the filtered basis functions are then selecto minimize tracking errors while satisfying the kinematic limitthe machine. The proposed method is shown to be very systemand easy to use, and it does not sacrifice tracking speed. Sectio

descresuperfproa

2. P

Finclufunc

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geneprevof nreprinteCNCknowoptipara

xdm

xdm

xdm

26664|fflfflfflfflfflfflffl

x

For

p1, .

basi

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Nj;0

whereprE + 1gene

gk ¼

Fig. 1

C. Okwudire et al. / CIRP Annals - Manufacturing Technology 65 (2016) 373–376374

ribes the method, while Section 3 presents experimentallts that demonstrate its superior tracking and contouringormance compared to Altintas and Khoshdarregi’s [9] ap-ch. Conclusions and future work are presented in Section 4.

roposed trajectory optimization method

ig. 1 shows the dynamics (G(s)) of a CNC feed drive axis,ding its unwanted vibration modes, given by the transfer

tion

¼XMi¼1

ai þ bis

s2 þ 2zivn;is þ v2n;i

(1)

re M represents the number of modes, while ai and bi are theue constants, zi is the damping ratio and vn,i is the natural

uency of the ith mode in rad/s. It is desired to generate x�dm, aified and optimized version of desired position trajectory xd,

that, when applied to the feed axis in place of xd, minimizestracking errors caused by G while satisfying the kinematicts of the machine axis.ig. 2 shows a block diagram of the proposed approach forrating x�dm; it is based on the concept of filtered basis functionsiouslyintroducedbytheauthorsinthecontextoftrackingcontrol

on-minimum phase systems [11]. Let xd = [xd(0) xd(1) . . . xd(E)]T

esent E + 1 discrete time steps of the x-component of a toolpathrpolated bya CNC (i.e., the desired trajectory).We assume that the

has look ahead capabilities such that the E + 1 time steps of xd arenin advance. Furthermore, weassume that the modified butun-

mized motion command, xdm = [xdm(0) xdm(1) . . . xdm(E)]T, ismeterized in the CNC using B-splines such that

ð0Þð1Þ...

ðEÞ

37775

{zfflfflfflfflfflfflffl}dm

¼

N0;mðj0Þ N1;mðj0Þ � � � Nn;mðj0ÞN0;mðj1Þ N1;mðj1Þ � � � Nn;mðj1Þ

..

. ...

} ...

N0;mðjEÞ N1;mðjEÞ � � � Nn;mðjEÞ

26664

37775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}N

p0

p1

..

.

pn

26664

37775

|fflfflffl{zfflfflffl}p

(2)

a B-spline of degree m, having n + 1 � E + 1 control points, p0,. ., pn, and knot vector, [g0 g1 . . . gm+n+1]T, Nj,m(j) are real-valueds functions given by [11,12]

ðjÞ ¼j�gj

gjþm�gj

Nj;m�1ðjÞ þgjþmþ1�j

gjþmþ1�gjþ1Njþ1;m�1ðjÞ

ðjÞ ¼ 1 gj�j < gjþ1

0 otherwise

� (3)

re j = 0, 1,. . ., n and j 2 [0, 1] is the spline parameter,esenting normalized time, which is discretized in Eq. (2) into

uniformly spaced points, j0, j1, . . ., jE. Without loss ofrality, the knot vector is selected to be uniform, such that

0 0�k�mk�m

n�m þ 1m þ 1�k�n

1 n þ 1�k�m þ n þ 1

8><>: (4)

Notice that if xdm given in Eq. (2) is applied as the motioncommand to the CNC feed axis of Fig. 1, in place of xd, the controlpoint vector, p, is not affected by G, because p is not a function oftime; however, the basis functions matrix, N, which is a function oftime, is filtered by G. Therefore, the resulting position x = [x(0) x(1). . . x(E)]T can be approximated by

x�Np (5)

e ¼ xd�x�xd�Np (6)

One useful mathematical property of B-splines is that thederivative of a B-spline curve is also a B-spline curve that canbe expressed linearly in terms of the control points of the originalcurve. Hence, the rth derivative of xdm can be written as [11]

drxdm

dtr ¼ Nrp (7)

where Nr is the basis function matrix for the rth derivative ofposition; its elements can be calculated readily from those of N[11,12]. Therefore, to achieve the stated objective of this paper,x�dm ¼ Np� can be generated using optimal control points, p*,calculated to minimize the objective function given by

p� ¼ arg minpðeTeÞ�arg min

pððxd�NpÞTðxd�NpÞÞ

s:t: kNrpk1�Lr

(8)

where Lr represents the kinematic limit associated with the rth

derivative of xdm (e.g., its velocity, acceleration and jerk limits),while k�k1 denotes the infinity norm (i.e., the maximum absolutevalue of the elements) of a vector.

Notice that, if the kinematic constraints are not considered, thesolution to Eq. (8) reduces to a traditional least-squares fittingproblem [11] where

p� ¼ ððNTNÞ�1

NTÞxd (9)

However, with the constraints included, Eq. (8) becomes a leastsquares problem with inequality constraints, which has aguaranteed solution as long as the constraints are feasible. Thesolution is unique if the matrix N is full rank (i.e., if the filteredbasis functions are linearly independent [11]). The authors haveshown in their previous work [11] that linear dependence of thefiltered basis functions is highly improbable. The proposed methodachieves trajectory parameterization, constraint enforcement andoptimization simultaneously using industry-standard B-splinesand basic least squares techniques, making it easy to use. Noticealso that both xd and x�dm have the same length (i.e., E + 1), meaningthat the proposed method does not sacrifice motion speed. Thoughthe method has been discussed for a single axis (i.e., x), it is readilyapplicable to multiple axes of a CNC machine by treating each axisseparately; therefore, it is very systematic and scalable.

3. Experimental validation

Experimental results are presented in this section to compare theproposed technique to the input shaping with contour errorcompensation (i.e., IS + CEC) technique proposed by Altintas andKhoshdarregi [9]. The biaxial (X–Y) linear motor driven stage(Aerotech ALS 25010) shown in Fig. 3 is used for the experiments.

. Dynamics G of a CNC feed axis including unwanted vibration modes.

Fig. 2. Block diagram of the proposed trajectory optimization method (applied to the x-component of a multi-axis toolpath).

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LPF),

C. Okwudire et al. / CIRP Annals - Manufacturing Technology 65 (2016) 373–376 375

The stage is controlled using a traditional P/PI feedback controller,augmented with velocity and acceleration feedforward [13]. Thecontroller is implemented on a dSPACE 1103 real-time control boardwith 10 kHz sampling frequency. The velocity and accelerationlimits for both of its axes are L1 = 100 mm/s and L2 = 8000 mm/s2,respectively. As shown in the Fig. 3, the stage is equipped with afixture consisting of a block mounted on a rod. The block is assumedto represent an apparatus (e.g., a tool, workpiece or measurementdevice) whose x and y positions are expected to track theirrespective desired trajectories, xd and yd, accurately, in spite ofinherent structural flexibilities. The acceleration of the fixture ismeasured using two unidirectional accelerometers (PCB Piezo-tronics 393B05) shown in Fig. 3.

Fig. 4 shows the frequency response function (FRF) of thedynamics G of each axis of the stage, generated by applying sweptsine acceleration (i.e., xd and yd) commands to the stage andmeasuring the corresponding accelerations of the fixture (i.e., x andy) using the accelerometers. The dynamics of each axis exhibitstwo dominant modes, which are very obvious in Fig. 4.

For each axis, the lower mode is a mode from the stage while thehigher mode is from the flexible fixture (whose natural frequenciesare different in the x and y directions because the mounting rod hasa rectangular cross section). However, not obvious in Fig. 4 are twomuch less dominant modes in each FRF. In other words, there arefour unwanted modes (M = 4) in each axis’ dynamics, whoseparameters are identified using the rational fraction polynomialmethod [14], and are summarized in Table 1. Fig. 4 shows anexcellent fit between the modeled and actual FRFs for the X-axis.

The model for the Y-axis is not as accurate as that of the X-because the two dominant modes of the Y axis are relatively mdamped and closer in resonance frequency, making it hardecurve fit them [15]. However, the fitting errors of the Y axis provan opportunity to test the performance of the proposed methothe presence of some modeling errors, which are likely to occupractice.

Fig. 5 shows the butterfly curve used as the desired path forexperiments (see [16] for details). Fig. 6 shows the positrajectories, xd and yd, (i.e., the uncompensated motion commanof the desired path, as well as its velocity and acceleration profias functions of time. Notice that the duration of the destrajectories is 1 s (i.e., E = 104, based on 10 kHz sampfrequency). The uncompensated trajectories are compared wthe corresponding modified trajectories obtained from IS + CECand from the proposed method. The zero-vibration and deriva(ZVD) input shaper is used for the IS + CEC method. For each mothe ZVD shaper consists of three impulses, A1, A2 and A3, placetimes 0, 0.5Td and Td, respectively, as reported in Table 2 (wheris the vibration period) [9]. Note that the shapers for the X anaxes are convolved so that both axes have the same shaper [9].

is performed using the method presented by Yang and Altin[17], which is superior to that presented by Altintas

Khoshdarregi [9]. Notice from Fig. 6 that when IS + CECimplemented without a low pass filter (LPF) applied to

modified motion commands, the axis velocity and acceleralimits are violated. This is because CEC introduces discontinui

Fig. 3. Experimental setup – biaxial stage with flexible fixture.

Fig. 4. Measured and modeled frequency response functions of X and Y axes of

experimental set up.

Fig. 5. Desired path.

Fig. 6. X and Y axis command signals for uncompensated, IS + CEC (without

IS + CEC + LPF and proposed (optimal) trajectories.

Table 1Model parameters of vibration dynamics for X and Y axes of stage.

Axis-mode# vn,i (Hz) zi ai bi

X-1 20.52 0.092 15,797.5 54.3

X-2 34.94 0.540 �135,160.6 �587.7

X-3 42.53 0.029 189,225.5 �60.5

X-4 42.60 0.007 14,633.4 �67.9

Y-1 17.86 0.120 6,709.0 310.4

Y-2 25.70 0.021 42,872.2 169.4

Y-3 30.66 0.440 �43,178.2 �1,260.2

Y-4 43.10 0.036 �966.3 7.5

Table 2Input shaper parameters.

Axis-mode# A1 A2 A3 Td

X-1 0.3273 0.4896 0.1831 0.0489

X-2 0.7787 0.2075 0.0138 0.0340

X-3 0.2733 0.4990 0.2277 0.0235

X-4 0.2554 0.4999 0.2446 0.0235

Y-1 0.3526 0.4824 0.1650 0.0564

Y-2 0.2668 0.4995 0.2338 0.0389

Y-3 0.6779 0.2909 0.0312 0.0363

Y-4 0.2791 0.4984 0.2225 0.0232

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C. Okwudire et al. / CIRP Annals - Manufacturing Technology 65 (2016) 373–376376

e trajectory [9]. Observe also that the resultant trajectories last47 s (i.e., 28.47%) longer than their uncompensated counter-s, due to the delays introduced by the input shapers. To deal

the discontinuities, constraint violation and excitation ofation modes due to CEC, a LPF with 3 Hz cut off frequency isied to the signals generated by IS + CEC, as recommended inNote, however, that there is no guarantee that constraints willys be satisfied after applying a specific (e.g., 3 Hz) LPF; thet of the LPF would vary depending on the given trajectory,

efore some iteration may be required. The proposed method isuted with B-splines of order m = 5, having n + 1 = 51 controlts. Notice from Fig. 6 that the modified trajectories generatedthe proposed method automatically satisfy the imposedmatic constraints and do not have any delays (i.e., they lasttly as long as the desired trajectories).he position trajectories from the uncompensated, IS + CEC + LPFthe proposed methods are applied to the X and Y axes of thee as motion commands; the IS + CEC trajectory without LPF isused in experiments because of its constraint violations. Fig. 7

s the fast Fourier transform (FFT) of x and y (measured using thelerometers) for the three methods. The X-axis accelerationgy spectral densities are 6.38 � 106 mm2/s4, 6.39 � 104 mm2/s4

1.92 � 106 mm2/s4, while the Y-axis acceleration energytral densities are 7.32 � 106 mm2/s4, 3.31 � 105 mm2/s4 and

6 2 4

are 1.0165 mm, 2.1709 mm and 0.1198 mm for the uncompensated,IS + CEC + LPF and the proposed methods, respectively. Fig. 9compares the contour errors of the three methods. Their RMSvalues are 0.5508 mm, 0.1852 mm and 0.0880 mm for theuncompensated, IS + CEC + LPF and the proposed methods, respec-tively, indicating the ability of the proposed method to indirectlyreduce contour errors via minimizing tracking errors. The implica-tion is that the proposed method does better than IS + CEC + LPF interms of the primary objective of improving tracking and contouringaccuracy, even if it means allowing a little more vibration.

4. Conclusion

This paper has presented a systematic, scalable and easy to usetrajectory optimization method that minimizes tracking errors inCNC machines that have unwanted vibration modes. The proposedmethod parameterizes the motion command to the machine as alinear combination of B-spline basis functions, which are thenforward filtered using a model of the machine’s dynamics. Theassociated control points are computed such that tracking errorsare minimized while ensuring that axis kinematic constraints arenot violated. The proposed method is compared in experiments toan input shaping and contour error compensation method, and isshown to yield significantly improved tracking and contouringaccuracy, without loss of productivity.

Acknowledgements

The authors would like to thank Mr. Deokkyun Yoon for hisassistance with the experimental setup presented in this paper.They would also like to thank Professor Hoshi for his constructivefeedback. This work is partially funded by the National ScienceFoundation’s CAREER Award #1350202.

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7. Fast Fourier transform (FFT) of measured acceleration of fixture for

mpensated, IS + CEC + LPF and proposed (optimal) trajectories.

. X and Y axis position tracking errors for uncompensated, IS + CEC + LPF and

sed (optimal) trajectories.

. Contour errors for uncompensated, IS + CEC + LPF and proposed (optimal)

tories.

� 10 mm /s for the uncompensated, IS + CEC + LPF and theosed methods, respectively. This means that the proposed

hod generates less vibration than the uncompensated; however,CEC + LPF provides the smallest level of vibration because inputing is focused on elimination of vibration while the proposed

hod focuses on accurate tracking – which are not exactly thee objective. To demonstrate this fact, the tracking errors, xd � x

yd � y, for all three cases are plotted in Fig. 8, based on positionals (x and y) derived from the measured acceleration signalsg an observer. The X-axis RMS tracking errors are 0.3014 mm,95 mm and 0.0852 mm, while the Y-axis RMS tracking errors

Annals of the CIRP 58(1):335–338.[14] Richardson M, Formenti D (1982) Parameter Estimation from Frequency

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