J. Software Engineering & Applications, 2009, 2: 288-294
doi:10.4236/jsea.2009.24037 Published Online November 2009 (http://www.SciRP.org/journal/jsea)
Adaptive Fuzzy Sliding Controller with Dynamic
Compensation for Multi-Axis Machining
Hu LIN 1, Rongli GAI2
1Shenyang Institute of Computing Technology, Chinese Academy of Sciences, Shenyang, China; 2School of Computer
Science and Technology, University of Science and Technology of China, Hefei, China.
Received June 31st, 2009; revised August 30th, 2009; accepted September 10th, 2009.
ABSTRACT
The precision of multi-axis machining is deeply influenced by the tracking error of multi-axis control system. Since the
multi-axis machine tools have nonlinear and time-varying behaviors, it is difficult to establish an accurate dynamic
model for multi-axis control system design. In this paper, a novel adaptive fuzzy sliding model controller with dynamic
compensation is proposed to reduce tracking error and to improve precision of multi-axis machining. The major ad-
vantage of this approach is to achieve a high following speed without overshooting while maintaining a continuous
CNC machine tool process. The adaptive fuzzy tuning rules are derived from a Lyapunov function to guarantee stability
of the control system. The experimental results on GJ-110 show that the proposed control scheme effectively minimizes
tracking errors of the CNC system with control performance surpassing that of a traditional PID controller.
Keywords: Fuzzy Sliding Control, Adaptive, Compensation Control, Tracking Error
1. Introduction
While basic machine tool errors form one of the major
sources of inaccuracy in multi-axis machining, achieving
high precision in actual machine tool performance also
critically depends upon dynamic performance of the in-
dividual axis controllers [1]. Axis controllers in process-
ing of CNC machining need to synchronize multi-axis
motions to generate the required machined surface [2]. In
general, each particular machine tool axis has its own
position and velocity, being driven separately along the
desired tool path generated by the interpolator of the
CNC system. When multi-axis machine tools are ma-
chining linear segments in “step” mode, the impact of
tracking error on machining accuracy is not a serious
problem. Otherwise, it would be difficult to eliminate
tracking errors while tracking the axial position com-
mand, which then contributes to contour error formation.
This situation would be especially problematic on the
conditions of continuous processing of the multi-axis
machining. As one example, Figure 1 provides an illus-
tration indicating the influence of tracking error in a
2-axis system while machining in the “auto” mode. The
interpolator is responsible for generating the ideal posi-
tion commands for coordinated movement of the axes,
which are indicated by positions Bi and Bi+1. Guided by
the interpolator, the axis controller manages the task of
controlling the movement of a particular axis. The actual
machining points of Ai and Ai+1 then correspond to in-
terpolation points Bi and Bi+1 due to the tracking error.
Accordingly, the tracking error is about |Bi-Ai| and
|Bi+1-Ai+1|, thus the machining error is |BD|, which is
called contour error, and is the result of the tracking error
of each axis.
Over the past several decades, many advanced con-
trollers have been designed which attempt to rectify these
tracking errors. Some examples of these include the
self-tuning fuzzy PID type controller [3−4], fuzzy sliding
controller [5], adaptive sliding controller with self-tuning
fuzzy compensation [6−10], adaptive neural fuzzy con-
trol [11], fuzzy controller and turning algorithm [12], and
self-tuning fuzzy logic controller [13].

Figure 1. The influence of tracking error

Adaptive Fuzzy Sliding Controller with Dynamic Compensation for Multi-Axis Machining 289
Overshooting and oscillation of the multi-axis control-
ler are special concerned in the article. Since it is difficult
to establish an accurate dynamic model of the multi-axis
machine tools, here, a novel adaptive fuzzy sliding model
controller with dynamic compensation is proposed. The
advantages of this model include its capacity to reduce
tracking error, thus contour error and improve multi-axis
machining precision.
2. Adaptive Fuzzy Sliding Controller with
Dynamic Compensation
Adaptive fuzzy sliding controller with dynamic compen-
sation is the core of the multi-axis CNC system. The ba-
sic function of this system is to generate the position of
servo axes of multi-axis machine tools. Digital encoder
feedbacks servo to monitor the position of the servo mo-
tor to the CNC system referred to as the actual axis loca-
tion of the machine tool. Interpolation of the CNC sys-
tem generates the ideal trajectory of each axis, which is
called the ideal location of axis of the machine tools.
While avoiding overshooting and oscillation in CNC
machining, the main role of the adaptive fuzzy sliding
controller with dynamic compensation is to minimize the
tracking error, as computed by the actual versus ideal
locations (Figure 2). The adaptive fuzzy sliding control-
ler with dynamic compensation is a steady-state control
system that includes two components - the dynamic
compensation and adaptive fuzzy sliding control.
2.1 Dynamic Compensation
Dynamic compensation is calculated from an array of
parameters including computing of tracking error, dead-
zone of tracking error, parameters of dynamic compensa-
tion and compensation control. The purpose of these de-
terminations is to minimize tracking error of the CNC
machining tools (Figure 3).
adaptive fuzzy sliding co nt ro ller
with dynam ic compens a ti o n
adaptive
fuzzy sliding
control DAC
encoder
amplifiers
hardwaresoftware
R+
C-
dynamic
compensation
motor
interpolation
points
Figure 2. The servo axis control system using adaptive fuzzy
sliding controller with dynamic compensation
Figure 3. The frame of dynamic compensation
2.1.1 Computing Tracking E r ror
The method used for computing tracking error consists of
calculating tracking error in each interpolation cycle
based on the actual versus ideal locations of the CNC
machining tool, as expressed in the following formula:
iii cpe 

(1)
where ),,,,,( CBAZYX iiiiiii eeeeeee is the tracking error in
the ith interpolation cycle, ),,,,,( iiiiiii cba
z
y
x
p
is the
position of the ideal location coordinate of the ith inter-
polation cycle, ),,,,,(iiiiiii cba
z
y
x
cis the position of the
actual location coordinate of the ith interpolation cycle,
as indicated by the digital encoder.
2.1.2 Deadzone of Tracking Error
The deadzone of tracking error parameter is used to de-
fine a range, in which dynamic compensation is not used.
The calculation of this parameter is based on the assump-
tion that the maximal distance between the actual loca-
tions of multi-axis and the corresponding interpolation
positions of multi-axis is within an interpolation step
when the program is running at the programming veloc-
ity. That is:
tvei

(2)
where ei is defined as presented in Equation (1), t

is
the interpolation cycle time of the multi-axis CNC sys-
tem, v is the programming velocity of the particular
workpiece program. It has been proven by Rogier that
the trajectory planner makes the maximal interpolation
error [14], expressed by emax , when it passes the adjacent
program with programming velocity. Since the maximal
interpolation error is accepted by the operator of CNC
system, it can be applied to bound the dynamic compen-
sation of tracking error, and be referred to as the dead-
zone of the tracking error. If, and only if, the tracking
error is greater than e
max, the dynamic compensation is
active. Practical experience shows that the pursuit of re-
ducing the tracking error to zero easily leads to vibration
and overshooting in multi-axis machining, so the intro-
duction of a deadzone of tracking error is conducive to
the stability of multi-axis controller.
2.1.3 Parameters of Dynamic Compensation
The overshooting of multi-axis machine tools will result
in oversize cutting. Since this is unacceptable for high
precision machining, dynamic compensation needs to
avoid overshooting and vibration of the multi-axis CNC
system. Parameters of dynamic compensation include the
following principles:
1) The parameters of dynamic compensation are zero
when the velocity of trajectory planning is zero.
2) The parameters of dynamic compensation are zero
when the tracking error achieves the constraint of the
deadzone, defined by chapter 2.1.2.
3) The parameters of dynamic compensation can in-

Adaptive Fuzzy Sliding Controller with Dynamic Compensation for Multi-Axis Machining 290
crease in a step-wise manner when the multi-axis CNC
system is running in an acceleration phase.
4) The parameters of dynamic compensation require a
step-wise reduction when the multi-axis CNC system is
running in a deceleration phase.
5) The parameters of dynamic compensation achieve a
peak value when the multi-axis CNC system is running
in the programming velocity.
Based on the principles of dynamic compensation, the
following function can be selected as the parameter of
dynamic compensation:
max
elee
v
v
comp ii
i
i (3)
Where vi is the velocity in the ith trajectory cycle, lei is
the length of the tracking error of multi-axis in the tra-
jectory cycle, defined by Equation (4), ,( X
ii compcomp
)
,,,, CBAZY iiiii compcompcompcompcompif the parameter of
dynamic compensation is in the trajectory cycle.
222222
CBAZYX iiiiiii eeeeeele  (4)
2.1.4 Compensation Contr ol
The output of the adaptive fuzzy sliding controller with
dynamic compensation for multi-axis machining pro-
posed in this paper can be expressed by the following
equation:
uuu ccol

 (5)
Where ucol is the total output of the position controller of
the multi-axis CNC system, ucis the output of the dy-
namic compensation controller and u is the output of the
adaptive fuzzy sliding controller. We can define the rules
for compensation control as follows:





max
max
00 eleu
elecompku
ic
iipc (6)
Where kp is the proportional gain of the CNC system,
while the meanings of other variables are presented as
Equation (3), (4) and (5).
The dynamic compensation controller adds a compen-
sation variable to each axis according to the vector of the
multi-axis tracking error. It can produce a rapid reduction
in the tracking error while simultaneously reducing the
frequency of adjusting parameters of the adaptive fuzzy
sliding controller. In addition, the dynamic compensation
controller can improve the stability of the numerical
multi-axis control system.
2.2 Adaptive Fuzzy Sliding
There exist a number of nonlinearities and uncertainties
in the multi-axis control system. These result from struc-
tural or unstructured uncertainties, such as backlash,
saturation and friction. It is very difficult to establish the
boundary for these nonlinearities and uncertainties in the
CNC system, particularly in a multi-axis system. Dy-
namic compensation parameters are changed according
to the trajectory velocity and therefore can contribute to
an increase in the uncertainty of the multi-axis system.
As an approach to rectify these problems, an adaptive
fuzzy sliding control structure is proposed. This structure
is referred to as adaptive fuzzy sliding controller with
dynamic compensation since it incorporated the dynamic
compensation algorithm discussed above.
Dynamic systems with multiple kinds of nonlinearities
and uncertainties, such as multi-axis machine systems,
can be expressed as the following Formula [15–17]:
() ˆ
()( ,)( ,)
n
x
tfXt fXtu

 (7)
Where (1)
( ,,.......,)
nT
Xxxx


 is the state of the system
and n
uR and n
x
R are the control inputs and out-
puts of the system, respectively. The nonlinear model
system consists of the reference model ˆ(,)
f
Xt and mul-
tiple nonlinearities and uncertainties (,)
f
Xt, which
include the dynamic compensation and inherent nonlin-
ear characteristics of the multi-axis CNC system. A hy-
pothesis can be proposed based on:
(,) (,)
f
xtF xt (8)
The time-varying sliding surface is defined as:
1
(,) 0.
n
d
sxt e
dt






 (9)
It is referred to as the sliding switching line in 2D
space, where λ is a positive constant and
Tn
deeeXXe ),...,,( )1( 
 is the tracking error vector.
In general, when considering second-order system as
an example, the ideal fuzzy sliding control law is:
),(),(),(
ˆ
*  ssatxxkextXfud  (10)
Where ),( xxk  and ),(

ssat are defined as:
(, )()kxxFx 
 (11)
1, 1
(, ),11
1, 1
s
sat sss
s

 





 



(12)
And η is a positive constant, while φ is the thickness of
the boundary layer.
Suppose:
12
12
2
1
2
11 1
()
()
((
li
i
li
i
i
A
i
mm
i
A
ll i
x
x
))
x




 

  (13)

Adaptive Fuzzy Sliding Controller with Dynamic Compensation for Multi-Axis Machining 291
1mm. Corners which were present between the small line
segments enabled for an easy detection of the tracking
error effect upon machining accuracy.
G61G05.1Q1F10000
X-1.513 Y215.223 Z-164.657 A86.462 C-91.007
X-1.426 Y216.091 Z-165.266 A86.556 C-91.002
X-1.339 Y216.975 Z-165.907 A86.659 C-91.992
X-1.253 Y217.874 Z-166.575 A86.77 C-90.976
X-1.166 Y218.784 Z-167.264 A86.887 C-90.956
X-1.08 Y219.706 Z-167.976 A87.011 C-90.929
X- .992 Y220.635 Z-168.701 A87.139 C-90.898
X- .905 Y221.572 Z-169.443 A87.272 C-90.86
X- 82 Y222.512 Z-170.194 A87.408 C-90.815
X- .733 Y223.458 Z-170.958 A87.548 C-90.765
X- .648 Y224.408 Z-171.732 A87.691 C-90.707
M2
3.1 Experimental Parameters
The parameter settings of the GJ-310: the axis number is
6, the encoder input equivalent is 16384, the servo peri-
odical time is 2ms, the interpolation cycle time is 2ms,
the maximal error is 0.2mm , the maximal shape error is
0.05mm, and the other parameters are shown as table 1.
According to established guidelines of adjusting nu-
merical control machines, we obtained the control rules,
as shown in table 2. Triangle membership functions of
the input variables are shown in Figure 5.
Figure 4. The program for test 3.2 Experiment Results
Where the adaptive law of fuzzy sliding controller can be
designed as [18]: When machining the program (Figure 4) with the GJ-310
CNC system, while separately using the adaptive fuzzy
sliding controller with dynamic compensation and the
2()
T
ep x  

＝ (14)
PID controller, we can get the position of each axis, as
shown in Figure 6. In Figure 6, series 1 is the ideal posi-
tion of each axis generated by the trajectory planner (in-
dicated with a dot), series 2 is the actual position of each
axis generated by the PID controller (indicated with a
dash) and series 3 is the actual position of each axis gen-
erated by the adaptive fuzzy sliding controller with dy-
namic compensation (indicated with a solid). We ob-
tained the tracking errors of each axis as shown in Figure
7, where series 1 is the tracking error generated by the
PID controller (indicated with a dot) and series 2 is the
tracking error generated by the adaptive fuzzy sliding
controller with dynamic compensation (indicated with a
solid). Note, the starting position of the program is indi-
cated by -1mm, 215mm, -150mm, 86mm, -91mm.
And γ is a positive quantity, P is a definite symmetric
matrix and P2 is the final array of the definite symmetric
matrix P. Stability of the adaptive fuzzy sliding control-
ler is guaranteed by the Lyapunov function.
3. Experiment
The experimental platform is a multi-axis CNC system,
referred to as GJ-310 CNC. This system is based on the
PC architecture, in which the servo board and the I/O
board are connected by the SSB bus. The adaptive fuzzy
sliding controller with dynamic compensation described
above is used as an axis position controller in the motion
control component of the GJ-310 CNC.
We selected a program with a simultaneous 5-axis
moving to test the proposed adaptive fuzzy sliding con-
troller as shown in Figure 4. The program consisted of a
micro-line segment, whose length was approximately
3.3 Experimental Analysis
From Figure 6 it is clear that the adaptive fuzzy sliding
Table 1. The parameters of GJ-310 CNC
category
Parameters axis X Axis Y Axis Z units Axis A Axis B Axis C units
D/A channel 1 2 3 5 6 8
Encoder channel 1 2 3 5 6 8
Proportional gain 33.33 33.33 33.33 33.33 33.33 33.33
Integral gain 0.1 0.1 0.1 0.1 0.1 0.1
Differential gain 0.01 0.01 0.01 0.01 0.01 0.01
Maximum velocity 30000 40000 40000 mm/min 20000 25000 30000 deg/min
Maximum error 15 15 15 mm 10 10 10 arcmm
Output offset 0.1099 0.2004 0.1061 mm 0.1796 0.2083 -0.1091 arcmm
Jog velocity 18000 18000 18000 mm/min 15000 15000 15000 deg/min
Transfer velocity 20000 20000 20000 mm/min 15000 17000 18000 deg/min
Maximum voltage 8 8 8 v 8 8 8 v
Jog acc-time 400 400 400 ms 600 600 600 ms
Transfer acc-time 400 400 400 ms 600 600 600 ms

Adaptive Fuzzy Sliding Controller with Dynamic Compensation for Multi-Axis Machining 292
Table 2. The control rules
Error change rate ∆e
The output u LN MN SN ZZ SP MPLP
LP LP LP LP LP LP LP LP
MP SP SP SP MP MP LP LP
SP ZZ SP SP SP MP MPLP
ZZ ZZ ZZ ZZ ZZ ZZ ZZZZ
SN LN MN MN SN SN SNZZ
MN LN LN MN MN SN SNSN
The
Tracking
Error
e
LN LN LN LN LN LN LNLN
Figure 5. The triangle membership functions of the input
variables
controller with dynamic compensation and the PID con-
troller produce nearly identical ending points. Moreover,
use of the adaptive fuzzy sliding controller with dynamic
compensation results in smooth control, thereby produc-
ing an accurate tracking of the trajectory with no over-
shooting or vibration in the machining.
From Figure 7 it is apparent that the adaptive fuzzy
sliding controller with dynamic compensation can reduce
tracking errors of each servo axis. As one example of the
program, an examination of axis Z which has the longest
moving trajectory, provides for a means of comparison of
the effects between the two controllers. The adaptive
fuzzy sliding controller with dynamic compensation re-
duced the tracking error by almost 44% of the PID con-
troller and the tracking error of axis C, whose moving
trajectory was the shortest in the example of program,
was reduced by about 19% of the PID controller.
From the above simulation results, the proposed adap-
tive fuzzy sliding controller with dynamic compensation
demonstrates a perfect performance, which can abolish
effects of the system trace performance caused by track-
ing nonlinearities and uncertainties disturbances. In this
way, the control system can produce a high degree of
precision in multi-axis machining.
4. Conclusions
Strictly speaking, the CNC system typically has more
than one axis. Therefore, an ideal linear system is not
available. In this way, research is directed at resolving
nonlinearity and uncertainty control problems of multi-
axis machining tools, and reducing tracking errors of
multi-axis. CNC systems have important implications for
Figure 6. The position of each axis

Adaptive Fuzzy Sliding Controller with Dynamic Compensation for Multi-Axis Machining 293
Figure 7. The tracking error of each axis
both theoretical considerations and practical applications.
This paper combines dynamic compensation control with
adaptive fuzzy sliding control for the design of an adap-
tive fuzzy sliding controller with dynamic compensation.
The results of experiment on the GJ-310 system show
that this controller can eliminate overshooting and vibra-
tion in a CNC machining control system, and improve
the precision of multi-axis machining.
5. Acknowledgment
This paper was supported by National Technology Sup-
port Program of Ministry of Science and Technology
under Grant 2007BAP20B01 and Chinese Academy of
Sciences Knowledge Innovation Program under Grant
KGCX2-YW-119.
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