An Analysis of X-Y Table Performance Via InputShaping

Shahrul Hamzah Abdul Razak1, Abdul Rashid Husain2, Zaharuddin Mohamed3, Mohamad Noh Ahmad4

Department of Control and Mechatronic Engineering

Universiti Teknologi Malaysia

81310 Skudai, Johor, Malaysia

shahrulhamzah85@yahoo.com1, rashid@fke.utm.my2, zahar@fke.utm.my3, noh@fke.utm.my4

978-1-4799-4653-2/14/$31.00 c© 2014 IEEE

Abstract—In industrial applications, vibration due to anundesirable non-linearities of a system such as friction causesa degradation of contouring performance in CNC machine. Inthis paper, an analysis of X-Y table performance for a class ofCNC system is carried out with the existence of friction. Aninput shaping is used to minimize the effect of vibration anda PID controller is used to achieve the desired performance ofX-Y table. The systems is configured to follow a varieties ofcontour profile to evaluate the system performance. The resultshows that the input shaping can greatly reduce the vibrationof the system.

Index Terms—Computer Numerical Control (CNC), Vibra-tion, Contouring.

Computer Numerical Control (CNC) has touched almost

every facet of manufacturing. Many machining processes

have been improved and enhanced through the use of CNC.

A good CNC machine has high contouring accuracy which

is determine as the ability of theCNC to produce a good

product. The contouring accuracy of the CNC machines is

affected by many factors, such as the geometrical inaccu-

racies of the machine’s axes of motion, inaccuracies in the

position feedback devices used, errors resulting from the

machining process, and in the imperfect control of the axes

of motion.

Friction at the sliding interface of the CNC machine work

table is resulted from the geometrical inaccuracies of the

machine’s axes. The friction appear in so call slip-stick

form. The stick friction mode is the resistance that against

the system to move at the beginning of the motion from their

static form while the slip friction mode is the resistance that

occursagainst the existing motion. The friction resistance is

a constrained force in the stick mode and an applied force

in the slip mode. When the work tables are in motion,

both phenomena are present, which resulting in a stick-

slip motion. Such responses also appear when there are

rapid motion of start and stops. In circular contours, friction

make the circular contouring accuracy degrade at quadrant

positions where there is reversal of velocities or motion from

standstill.

Various research work has been done on modeling and

control of system with friction. Early researches contribute

the mathematical model of the dynamics of the friction.

In [1], a new model of system with friction is investigate

experimentally. This includes the Stribeck effect, hysteresis,

spring-like characteristics for stiction, and varying break-

away force. A new friction observer is designed based on

that characteristics. In [2], a dynamic servo model has been

combined with a friction model is controlled with trape-

zoidal velocity control algorithm due to friction dependency

of velocity. A compensation of static friction during the

transition from the presliding regime to the sliding regime

is proposed in [3].

In machine tools, when the work table moving at a slow

speed, the system may execute jerky motion or vibration in-

stead of a smooth travel. In milling process, a vibration can

occur when the vibration can be one of the factors that limit

the productivity. The consequences of this phenomenon

is poor quality of surface, higher cutting forces implying

higher wear of the tool or even tool breakage [4]. In [5],

a vibration is reduced by using input shaping method. The

input shaping proposed can avoid triggering the excitation of

structural vibration frequencies. A vibration avoidance using

an input shaping during high speed machining is proposed

in [6]. The method is effective in single axis positioning

but increase contouring error in multi-axis application. [7]

proposed both vibration avoidance by using same method

but with contouring compensator for multi-axis machine

tools.

This work focuses on friction and vibration analysis on X-

Y table. An input shaping controller is utilized to minimize

the vibration of the system. Furthemore, a PID controller

is used to obtain the desired performance. The system is

tested with several input profile to observe the response and

the effectiveness of the input shaping used. The control of

contouring error and tracking error is not included in this

framework. The result is based on the fixed setting of PID

controller.

This paper is organized as follows; Section I describes

the mathematical model of the X-Y table and friction used

in this analysis. Section II explains about zero vibration

derivative and derivative (ZVDD) shaper algorithm. Section

III discusses the configuration of X-Y table system and

simulation results. Section IV is the conclusion of this

reported work.

I. MATHEMATICAL MODEL OF X-Y TABLE

The X-Y table system involves linear axis X and linear

axis Y. For controller design purpose, only the dynamic

model of X-Y table without stepper motor model is con-

sidered. The equation of motion X table and Y table are

given by equations (1) and (2) respectively:

Mxx+ Cxx+Kxx =Fx +Mxgµ(V0x + x) + V0x (1)

My y + Cy y +Kyy =Fy +Mygµ(V0y + y) + V0y (2)

where x and y, x and y, y and y are position, velocity,

and acceleration of X and Y axes, respectively. Mi, Ci and

Ki are the mass, damping, and stiffness of the system. gis gravitational acceleration and Fx and Fy are the control

input forces. V0i and V0i are the velocity and acceleration

of input force. Friction model is included to the system

where µ is the coefficient of friction as shown in eq. (3).

The coefficient of friction is a modified version of that of

[8] and reflects a falling and rising characteristic typical of

machine tools.

µ(v) = [µ1v2 + (µ1γ − µ2)sechβv]tanhαv (3)

where µ1 is the static coefficient of friction and µ2 is the

dynamic coefficient of friction. v is the non-dimensional

relative velocity between the slider and the guide way. α,

β and γ are fitting parameters.

−1 −0.5 0 0.5 1

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Relative velocity

Co

effi

cien

t o

f F

rict

ion

Fig. 1. Coefficient of friction vs Relative velocity

Fig. 1 shows the coefficient of friction response with

relative velocity. At the begining of the motion, the axis

cannot move until it breaks the static friction force. After

that the friction force decrease and increase acording to the

relative velocity.

II. INPUT SHAPING CONTROLLER

Many input shapers are designed to overcome the vibra-

tion such as a zero vibration (ZV) shaper and zero vibration

derivative (ZVD). In this paper a zero vibration derivative

and derivative (ZVDD) shaper is used. A ZVDD is consist

of four impulses applied at tj = 0, 0.5Td, Td, 1.5Td, which

is the commonly used for real application as it results in

zero vibration and due to its robustness against the change

in the natural frequency [9], [10]. The impulse amplitudes,

Aj can be identified from equations (4) and (5).

A1

A2

A3

A4

=

1 1 1 1K3 −K2 K −10 −K2 2K −30 −K2 4K −9

−1

1000

=1

K3 + 3K2 + 3K + 1

13K3K2

K3

(4)

where K = e−Πζ

/√1−ζ2

and Td = 2Π/√

1− ζ2.

[

{

At

}

j

]

=

[{

A1

0

}

1

,

{

A2

0.5Td

}

2

,

{

A3

Td

}

3

,

{

A4

1.5Td

}

4

]

(5)

where j = 1, 2, 3, 4Fig. 2 shows the effect of ZVDD shaper to the input

reference versus time.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Time (s)

Am

pli

tud

e

Fig. 2. ZVDD Input Shaper response

III. SIMULATION RESULTS AND DISCUSSION

In order to simulate the performance of X-Y table with

input shaping controller, the system is designed to have

chattering by default. Fig. 3 shows the schematics block

diagram for the X-Y table. In this simulation, minimizing

the contouring error is not the main concern. Therefore, the

PID is tuned to have a settling time (Ts = 0.1s) to make

sure the system is stable and able to achieve steady state

error ess close to zero. The system is tested with a square,

zigzag and circle contour profile.

Fig. 3. Block diagram of X-Y table

Simulation is performed by using Matlab/Simulink-based

simulator. The following parameters value have been con-

sidered for simulation. Mx = 190kg, My = 220kg,

Cx = 1900N− s/m, Cy = 2200N− s/m, Kx = 108N/m

and Ky = 108N/m. The damping value is based on the

damping ratio of 1% of the critical damping calculated

from the stiffness data. The numerical value for coefficient

of friction, µ in equation (3) are µ1 = 0.15, µ1 = 0.08,

α = 50s2/m2, β = 5s/m and γ = 1.5.

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

X−axis position(m)

Y−

axis

po

siti

on

(m)

Reference

Unshaped

Shaped

Fig. 4. The desired circle profiles generated by interpolator

Fig. 5. The actual X-axis position (Circle)

Fig. 6. The actual Y-axis position (Circle)

In the first case, a contour of full circular motion is tested

to examine effectiveness of the input shaping. Fig. 4, Fig. 5

and Fig. 6 shows that the contouring tracking for reference

input, unshaped output and shaped output. Unfortunately,

there is no vibration found in the result but the shaped output

response is slower than the unshaped output. There is a

tracking error between input and output in both shaped and

unshaped output due to friction acting on the axis.

Fig. 7. The desired square profiles generated by interpolator

Fig. 8. The actual X-axis position (Square)

Fig. 9. The actual Y-axis position (Square)

The second case is a contour of square motion. Fig. 7

shows that the desired square profile of X-Y table. In each

corner of the square motion, vibration occur when there is

an interchanging motion between axis. From Fig. 8 and Fig.

9, it seems that the input shaping is successfully eliminate

the vibration. The vibration of unshaped output form a circle

along the contour as presented in Fig. 7.

Fig. 10. The desired Zigzag profiles generated by interpolator

Fig. 11. The actual X-axis position (Zigzag)

Fig. 12. The actual Y-axis position (Zigzag)

The third case, a contour of zigzag motion is tracked to

examine the effectiveness of the input shaping as shown

in Fig. 10. The vibration in both Fig. 11 and Fig. 12 is

eliminated but the input shaping introduce huge error to the

contour. The unshaped output seems to follow close to the

reference input but the vibration become big before each

axis changing direction.

IV. CONCLUSION

This article shows that the X-Y table is implemented with

input shaping controller and PID controller. Based on the

simulation, the input shaping shows a good performance

in eliminating the vibration of the system but it introduces

a delay to the system. Further improvement is needed to

minimized the contouring error of the system. It is also

shown that the vibration appears when a rapid motion of

input is given to the system.

ACKNOWLEDGMENT

The authors would like to thank the MOSTI and

Universiti Teknologi Malaysia (UTM) for SF Vote No:

R.J130000.7908.4S070 grants that have been supporting this

research financially.

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