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1. SCIENCE CHINA
Technological Sciences
© Science China Press and Springer-Verlag Berlin Heidelberg 2015 tech.scichina.com link.springer.com
*Corresponding author (email: chen_wei@tju.edu.cn)
• Article • April 2016 Vol.59 No.4: 680–688
doi: 10.1007/s11431-015-5988-8
Robust adaptive cross-coupling position control of biaxial
motion system
CHEN Wei1*
, WANG DianDian1
, GENG Qiang2
& XIA ChangLiang1,2
1
School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China;
2
Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, Tianjin Polytechnic University,
Tianjin 300387, China
Received September 23, 2015; accepted November 20, 2015; published online December 25, 2015
The stability and synchronous performance are usually hard to be improved simultaneously in the biaxial cross-coupling posi-
tion motion control system. Based on analyzing the characteristics of the cross-coupling control system, a robust adaptive
cross-coupling control strategy is proposed. To restrict influences of destabilizing factors and improve both of stability and
synchronous performance, the strategy forces dual axes to track the same reference model using Narendra adaptive control
theory. And then, a robust parameters adaptive law is proposed. The stability analysis of the proposed strategy is conducted by
applying Lyapunov stability theory. Related simulations and experiments indicate that the proposed strategy can improve syn-
chronous performance and stability simultaneously.
biaxial motion, motor control system, position tracking, robust adaptive control, cross-coupling, synchronous error
Citation: Chen W, Wang D D, Geng Q, et al. Robust adaptive cross-coupling position control of biaxial motion system. Sci China Tech Sci, 2016, 59:
680688, doi: 10.1007/s11431-015-5988-8
1 Introduction
The biaxial motion system has found numerous applications
in mechatronics occasions, including precision machining,
spinning, steel rolling and so on [1–3]. In a biaxial system, a
couple of servosystems are usually linked by a sliding table
or other mechanical structure, and control the object to track
the position trajectory through synchronous motion. During
the working process, a biaxial synchronous error occurs
because of the unbalanced load, mechanical assembly error
and uncertain disturbances. The synchronous error will re-
duce the quality of products or even damage the equipment
[4]. Hence, the research of high-performance biaxial control
strategies has practical significance and necessity.
In these years, efforts to reduce synchronous error have
been made by improving the tracking accuracy of each axis
individually. Tomizuka proposed a zero phase-error track-
ing control strategy (ZPETC) [5], in which a feedforward
controller is constructed based on the reciprocal of system
transfer function. Several modified ZPETCs have subse-
quently been proposed to reduce tracking error [6,7]. Refs.
[8–10] design feedback and feedforward controllers for indi-
vidual axis, and all of them can improve tracking perfor-
mance effectively. However, because of the servo lag, a
good tracking performance for each individual axis does not
guarantee further reduction of biaxial synchronous error
[11,12].
To solve this problem, Koren introduced a cross-coupling
control (CCC) strategy [13]. It considers the mutual dy-
namic effects of dual axes to further reduce the synchronous
error. Various CCCs were then proposed for dual axes by
applying robust control [14], adaptive control [15], sliding-
mode control [16], iterative learning control [17], predictive

2. Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4 681
control [18] and fuzzy control [19] to improve the syn-
chronous accuracy.
The stability and synchronous performance of biaxial
CCC system are hard to be balanced. Aiming at this prob-
lem, this paper proposes a robust adaptive cross coupling
control (RACCC) strategy. The property of CCC system is
fully analyzed, and then a synchronous error converting
system (SECS) is established to study the destabilizing fac-
tors in biaxial system. Based on Narendra adaptive control
theory, RACCC strategy designs a reference model and a
robust adaptive law for dual axes to suppress these destabi-
lizing factors. Thus the stability and synchronous perfor-
mance can both be improved. The stability analysis of pro-
posed strategy is conducted using Lyapunov stability theory
in Appendix A. Finally, simulations and experiments are
performed and some results are illustrated to validate the
effectiveness of RACCC strategy.
2 Synchronous error of biaxial system
In biaxial system, the outputs of motor systems are rotor
positions. Two motors are linked by a sliding table or other
mechanical structure to convert rotor positions into the dis-
placement of X axis and Y axis. And then, planar motion of
the object can be achieved.
Figure 1 shows the model of synchronous error. P*
=[P1
*
P2
*
]T
is reference position of the object; P=[P1 P2]T
is actual
position; 0 is the angle of position trajectory; e1, e2 are
tracking errors;  is synchronous error.
Under ideal conditions, the object can track reference
trajectory accurately. However, due to the limit of control
accuracy and uncertain disturbances, actual position trajec-
tory of the object usually deviates from reference trajectory.
According to Figure 1, the tracking error and synchronous
error can be defined as
 
T
T
1 2 1 1 2 2
e e P P P P
 
 
     
 
e P P

, (1)
1 0 2 0
sin cos
e e
  
    
L e , (2)
where L=[sin0 cos0] is a transfer matrix.
Figure 1 Model of synchronous error.
3 Biaxial CCC system
3.1 Synchronous error of CCC system
In biaxial CCC system, the tracking errors of individual axis
are changed into synchronous error and then be used to
modify the position command. Biaxial synchronous error
can be reduced by considering the mutual effects of both
axes. Figure 2(a) shows the block diagram of CCC system.
In Figure 2(a), 1
*
, 1 and 2
*
, 2 are the input and output
speed signals of motor control systems in X axis and Y axis,
respectively. c is synchronous error. C is coupling control-
ler; Gp1(s), F1(s), Gp2(s) and F2(s) are position feedback and
feedforward controller of X axis and Y axis, respectively.
The individual axis system employs typical servo system as
shown in Figure 2(b), which consists of position feedback
controller, feedforward controller and motor speed control
system.
In Figure 2(b), J, B and Kt are rotational inertia, viscous
coefficient and torque constant, respectively. The motor
speed control system is a traditional double closed-loop PI
system. Considering the fast response of current loop, its
transfer function can be approximated by 1. The velocity
controller is of PI type, where Gv1(s)=kv+ki/s. The expres-
sions of feedback controller and feedforward controller are
Gp1(s)=kp and F1(s)=s. Thus the closed-loop transfer func-
tion of X axis system can be derived as
2
v i p v p i
1
3 2
v i p v p i
t t
( )
( ) .
( )
k s k k k s k k
M s
J B
s k s k k k s k k
K K
  

 
    
 
 
(3)
Y axis employs the same control structure as X axis. So
the nominal models of Y axis and X axis are the same. Only
when the parameters of their motors or controllers are dif-
ferent, the model coefficients have differences.
Figure 2 Cross coupling control system.

3. 682 Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4
Figure 3 shows the simplified block diagrams of biaxial
system with and without cross coupling part. The Figure 3(a)
and (b) is uncoupled system and CCC system, respectively.
In Figure 3, M=diag (M1, M2); o is the synchronous error of
uncoupled system; coupling controller C is of P type, where
C=kc.
The tracking error and synchronous error of uncoupled
system can be expressed as
( )
 
e I M P
, (4)
o ( )
  
L I M P
, (5)
where I is a unit matrix.
The synchronous error in CCC system can be derived as
T 1
c c
( ) ( )
k
 
  
L I ML L I M P
. (6)
The matrix inversion lemma is applied in
T 1 T 1 T
c c c
( ) ( )
k k k
 
   
I ML L I M I L LM L L . (7)
Substituting eq. (5) and eq. (7) into eq. (6) yields
T 1 T
c c c o
[ ( ) ]
k k
 

  
I LM I L LM L . (8)
Applying the matrix inversion lemma again, eq. (8) can
be simplified as
c o
 
 
F , (9)
where F=(1 + LMLT
kc)1
.
Eq. (9) depicts the relation of synchronous errors be-
tween the coupled and uncoupled system. F is defined as
synchronous error transfer function which can be used in
performance analysis as a sensitivity function of CCC sys-
tem. If the gain of F is decreased, the c will reduce along
with it.
3.2 Performance analysis of CCC system
In CCC system, synchronous performance can be improved
by decreasing the gain of F through the proper choice of
controller C. However, the stability of biaxial system often
decreases at the same time. Effects of controller C on CCC
system are discussed as follows.
The actual math models of X axis and Y axis can be re-
spectively expressed as
1 1 1
[1 ( )]
M M s

   , (10)
Figure 3 Control system diagram.
2 2 2
[1 ( )]
M M s

   , (11)
and let
2 1 3
[1 ( )]
M M s

  , (12)
where 1(s), 2(s) and 3(s) are rational fractions. 1(s) and
2(s) include model errors and uncertain disturbances of X
axis and Y axis, respectively. 3(s) represents the model
differences between dual axes when their motors or con-
trollers are different.
Eq. (3) can be simplified as
2
2 1 0
1 3 2
2 1 0
b s b s b
M
s a s a s a
 

  
. (13)
Replacing the nominal models M1 and M2 with actual
models 1
M  and 2
M , from eqs. (10)–(12), we have
T
1
= [1 ( )]
M s


LML , (14)
where (s)=[2(s)+3(s)+2(s)3(s)]cos2
0+1(s)sin2
0.
0 is decided by the input position trajectory. Thus, (s)
is determined only by 1(s), 2(s) and 3(s).
Substituting eq. (14) into eq. (9) yields
 
3 2
2 1 0
c o
3 2
2 2 1 1 0 0
( ) ( )
s a s a s a
s a Db s a Db s a Db
 
  

     
, (15)
where
c[1 ( )]
D k s
   . (16)
Eq. (9) can be regarded as a control system whose input
is o and output is c. The system can be defined as a syn-
chronous error converting system (SECS). Its transfer func-
tion is F, controller is D. Thus, the design of C in CCC sys-
tem is equivalent to design of D in SECS. The design goals
are to reduce c and keep SECS stabilized simultaneously.
Effects of controller D on SECS are discussed as follows.
3.2.1 (s)=0
If (s)=0, from eq. (16), we have D=kc=C. The routh chart
of SECS can be listed as shown in Table 1.
According to routh criterion, for SECS, the necessary
and sufficient condition of its stability is that all the ele-
ments in the first column of Table 1 are positive. As the pa-
rameters of motors and controllers in Figure 2 are positive,
Table 1 Routh chart of SECS
First column Second column
s3
1 a1+Db1
s2
a2+Db2 a0+Db0
s1
(a1a2+Da2b1+Da1b2+D2
b1b2–a0–Db0)/(a0+Db2)
s0
a0+Db0

4. Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4 683
the coefficients of M1 and D are certainly positive. Thus the
elements in the first, second and forth lines are all positive.
We only need to discuss the element in the third line.
From routh criterion, if X axis system can run steadily,
the coefficients of M1 should satisfy the following rule
0 1 2
a a a
 . (17)
By comparing eq. (3) with eq. (13), the following equa-
tions can be obtained: a0=b0, a1=b1. Combining them with
eq. (17), we have
2
1 2 2 1 1 2 1 2 0 0
0 2
0
a a Da b Da b D b b a Db
a Db
    


. (18)
Eq. (18) always holds which means that SECS is steady
whatever the parameter of controller D is.
According to the final value theorem and eq. (15), the
steady-state output of SECS is derived as
3 2
2 1 0
3 2
0
2 2 1 1 0 0
c
1
lim
( ) ( )
1 1
.
1 1

  
 
     
 
 
s
s a s a s a
s
s s a Db s a Db s a Db
D k
(19)
It can be seen the larger D is, the smaller c is. Because
the value of D has no effect on the stability of SECS, in-
creasing kc can improve synchronous performance effec-
tively without deteriorating the stability of CCC system.
3.2.2 (s)≠0
When (s)≠0, we have D≠kc. It’s not difficult to find from
eq. (16) that (s) is included in controller D and the effect
of (s) on SECS increases with the increase of kc. Thus, the
stability of SECS is hard to be guaranteed and the conclu-
sion above is no longer correct. This means the increasing
of kc has unpredictable effect on CCC system, especially on
synchronous error transfer function F. The result is that
steady operation of CCC system cannot be guaranteed.
In practical applications, the assumption (s)=0 usually
cannot be satisfied. The nominal models of X axis and Y
axis differ when their motors or controllers are not the same
totally. On the other hand, the model errors and uncertain
disturbances cannot be ignored either. The reason why sta-
bility and synchronous performance of CCC system are
hard to be improved simultaneously is that the effect of (s)
becomes more serious with the increase of kc.
Due to the existence of (s), the value of kc has to be de-
creased in order to guarantee steady operation of the system
with the sacrifice of synchronous performance. Thus, the
effect of (s) must be suppressed in order to achieve better
synchronous performance.
4 Robust adaptive CCC system
Narendra adaptive control theory is a model reference adap-
tive control scheme which neednot the precise math model
of controlled system. It can generate adaptive control varia-
ble only based on the input and output signals [20,21].
As can be seen in eq. (14), (s) is composed of 1(s),
2(s) and 3(s). In practical applications, the position feed-
back and feedforward controller of X axis and Y axis are
usually the same. That is to say Gp1(s)=Gp2(s), F1(s)=F2(s).
So the model differences lie mainly between the motor
speed control systems of X axis and Y axis. 1(s) and 2(s)
can also be equivalent to model errors and uncertain dis-
turbances of motor speed control systems.
The RACCC strategy retains the cross coupling part and
designs speed control system for individual axis as shown in
Figure 4. To suppress model differences, the motor speed
control systems of dual axes with different nominal models
are forced to track the same reference model. A robust pa-
rameters adaptive law is designed. It can make sure that the
controlled system can track reference model accurately even
with the existence of model errors and uncertain disturb-
ances.
The robust adaptive speed control system of X axis is
shown in Figure 4. The shaded part is double closed-loop
system whose nominal model is G0(s). Gm(s) is reference
model; m is the output of Gm(s); e is the difference
between m and 1; k0, d1, d2, d0 are all adjustable parame-
ters; F1 and F2 are auxiliary signal generators.
By applying Narendra adaptive theory, the adaptive con-
trol variable u can be generated based on input and output
signals of the double closed-loop system. The design pro-
cedures are as follows.
1) G0(s) can be expressed as
v t i v
0 2
v t i t
/
( )
( )


  
k k s k k
G s
J s B k k s J k k J
. (20)
Gm(s) should have the same structure as G0(s)
m
m m m 2
m 1 0
( )
( )
( )
N s s n
G s k k
D s s h s h

 
 
, (21)
where km is the gain of Gm(s), Nm(s) and Dm(s) are both
monic polynomials, Gm(s) must be a positive real function.
From the definition of positive real function, coefficients
of Gm(s) should satisfy the following rule.
Figure 4 X axis robust adaptive speed control system.

5. 684 Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4
1
h n
 . (22)
As Gm(s) and G0(s) have the same structure, we have
m 0
k n h
 . (23)
The bandwidth and damping coefficient of Gm(s) can be
obtained as
0
1 0
,
/ 2 ,
f n h
h h

  





(24)
where f and  need to be designed according to dynamic
performance requirements.
Eqs. (22)–(24) are the constraint conditions of Gm(s).
2) The state equations of F1 and F2 can be established as
1 1
v v u
 
 G g , (25)
2 2 1
v v 
 
 G g , (26)
where G and g are first order matrices, v1 and v2 areauxilia-
ry signals. Eqs. (25)–(26) should satisfy: sG=Nm(s).
When =[1
*
v1 v2 1]T
, =[k0 d1 d2 d0]T
, the adaptive
control variable u can be determined as
T
0 1 1 2 2 0 1=
u k d v d v d
 

      . (27)
3) The model error and uncertain disturbances of double
closed-loop system can be represented as 1(s).  is a pa-
rameter. The actual model of motor speed control system
Gp(s) can be defined as
p 0 1
( ) ( )[1 ( )]
G s G s s

  . (28)
From eq. (10) and Figure 2(b), the relation between 1(s)
and 1(s) can be described as
1
1
p 0 1
( )
( )
( )[1 ( )]
s s
s
s k G s s




 
. (29)
It can be seen in eq. (29), the suppression of 1(s) is
equivalent to suppression of 1(s). The design goal of
adaptive law is that the double closed-loop system can track
Gm(s) accurately even if 1(s)≠0.
Due to e=1m, the adaptive law can be designed as
0 c 0
0 c 0 0 c 1
0 1 0 c 1
( )
( )
e
E
k E E E
k E E E
 
 
   
 
    
   


      


    



  
(30)
where , 0 and k are positive real numbers,  is positive
definite diagonal matrix,  is a correction parameter, E0 and
E1 are threshold values.
The steps above compose the design procedure of robust
adaptive speed control system of X axis. The speed control
system of Y axis is the same as the one of X axis in order to
achieve the suppression of 3(s).
Eq. (30) adds a correction term to the adaptive law in
Narendra adaptive theory. This is because the original adap-
tive law is an integration adaptive law based on Lyapunov
stability theory. If 1(s)≠0, the negative definite of Lya-
punov function’s derivative cannot be guaranteed. To make
 approach desired value and guarantee the steady operation
of system, it is necessary to add a correction term . If
1(s) is small, a big correction term will lead to steady
state error between the output of Gp(s) and Gm(s). To avoid
this phenomenon, the correction term should be adjusted
according to c. It should stay smaller to reduce the steady
state error when 1(s) is small, and become bigger to add
the stability margin when 1(s) is big. The dual axes both
employ this adaptive law to restrict 1(s) and 2(s).
To sum up, RACCC strategy suppresses the effects of
(s) on biaxial system from the root. Steady operation of
biaxial system can be guaranteed even if kc increases. Thus,
the stability and synchronous performance can be improved
simultaneously. The stability analysis of RACCC is shown
in Appendix A.
5 Simulation and experiment analysis
5.1 Simulation analysis
In order to research the performance of RACCC with dif-
ferent coupling controller gain kc, the proposed system has
been implemented in the Matlab/Simulink programming
environment. The parameters of PMSMs in biaxial system
are given in Table B1 in Appendix B. In the simulation,
reference trajectory is a ramp signal whose angle 0 is 45°.
Feed rates of X axis and Y axis are both 100 mm/s. The
bandwidth and damping coefficient of Gm(s) are 100 rad/s
and 1, respectively. The motors started with no load at first.
Load torque of Y axis steps from 0 to 5 Nm at 2 s. Syn-
chronous performance and stability of biaxial system with
load disturbance can be tested. Coupling controller gain kc is
chosen as 20, 40 and 80. The corresponding simulation re-
sults are shown as Figure 5, respectively.
Figure 5(a) and (b) is the simulation results of CCC and
RACCC strategy when kc=20. As shown in the output tra-
jectory waveform, when system starts up and load steps,
synchronous error of RACCC system is smaller than CCC
system. The synchronous performance of biaxial system is
improved effectively by RACCC. From the position track-
ing responses of individual axis we can see that the tracking
error of RACCC system is decreased when load steps. The
reason is that RACCC strategy provides sufficient stability
margin for system to overcome uncertain disturbance.
Figure 5(c) and (d) is the simulation results when kc=40.
Compared with the results when kc=20, the synchronous
errors of CCC and RACCC system are both decreased. It

6. Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4 685
Figure 5 (Color online) Cross coupling controller kc=20: (a) CCC; (b) RACCC. kc=40: (c) CCC; (d) RACCC. kc=80: (e) CCC; (f) RACCC.
proves that synchronous error will decrease as coupling
controller gain kc increases.
Figure 5(e) and (f) shows the simulation results when
kc=80. As synchronous error waveforms of CCC system
depicted, the synchronous performance is deteriorated. Due
to the influences of (s), drastic position fluctuation appears
in CCC system which leads to the instability of dual axes. It
means that the CCC strategy can no longer guarantee the
operation performance of biaxial system when kc=80. In
contrast, RACCC system can still run steady and synchro-
nous error c is further decreased than Figure 5(a)–(d).
In order to illustrate the effect of coupling controller gain
kc, we conduct a histogram to describe the simulation results
above. Figure 6 depicts the synchronous errors of CCC and
RACCC system with different kc when system starts up and
load steps.

7. 686 Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4
In Figure 6, we can clearly find that c of both CCC and
RACCC system decrease along with the increasing of kc.
The phenomenon coincides with the theoretical analysis.
In conclusion, as said in Section 3 of manuscript, in-
creasing coupling controller gain kc is beneficial to decreas-
ing c. However, as kc increases to some extent, CCC strat-
egy can no longer guarantee steady state performance. In
contrast, whatever the value of kc, the c of RACCC system
is always less than CCC system. The comparison of per-
formance measures verifies the favorable control perfor-
mance of RACCC strategy clearly.
5.2 Experiment analysis
To test the feasibility and validity of RACCC, a biaxial po-
sition control experiment platform is set up as shown in
Figure 7. There are two permanent magnet synchronous
motors (PMSMs) driving mechanical arms to achieve planar
motion. The parameters are given in Table B1 in Appendix
B. The control chip is chosen to be TMS320F28335 pro-
duced by TI, whose clock frequency being 150 MHz.
Switching frequency of the IPM is 5 kHz. The sampling
time of the control loop is 200 s.
5.2.1 Synchronous performance comparison experiment
In this experiment, the reference trajectory is a broken line,
whose initial angle 0 is 45°. The trajectory turns at 6 s, 0
becomes 45°. The feed rate is 125 mm/s. The bandwidth
and damping coefficient of Gm(s) are 100 rad/s and 1, re-
spectively. The motors started with no load at first. Load
Figure 6 Histogram of simulation results. (a) Start up; (b) load step.
Figure 7 (Color online) Biaxial position control experiment platform.
Figure 8 Cross coupling control. (a) Position trajectory; (b) synchronous
error; (c) position of X axis; (d) position of Y axis; (e) tracking errors.
torque of X axis steps from 0 to 5 Nm at 2 s. The load re-
duces to 0 Nm at 8 s. Synchronous performance of biaxial
system with load disturbance can be tested. Figure 6 shows
the experimental results of CCC strategy when kc=40.
It can be seen from Figure 8 that synchronous error c is
2.8 mm when system starts up. When the load torque in-
creases suddenly at 2 s, c is 2.2 mm. At 6 s, the trajectory
turns and c becomes 4.5 mm. When the load torque steps

8. Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4 687
back to zero at 8 s, c is 2.8 mm. The tracking responses of
X axis and Y axis are depicted as Figure 8(c) and (d). Figure
8(e) shows the tracking errors of individual axis. It can be
seen that tracking errors are sensitive to load disturbance
and hard to approach zero in a short time.
If kc is increased to 60, the CCC system will close down
because of overcurrent protection. It means that CCC sys-
tem cannot operate steadily when kc=60. Compared with
CCC system, RACCC system still can operate steadily
when kc=80, the experimental results are shown in Figure 9.
In Figure 9, synchronous error c becomes 1.5 mm
when system starts up. c is 0.2, 1.7 and 0.2 mm at 2, 6,
8 s, respectively. Tracking errors are robust to load disturb-
ance and can approach zero in a certain time.
By comparing Figure 9 with Figure 8, we can see that
RACCC strategy can effectively improve synchronous per-
formance of biaxial system, especially when load disturb-
ance appears. From the position tracking responses of indi-
vidual axis, tracking errors are reduced and can rapidly ap-
proach zero in RACCC system. What’s more, because of
the suppression of (s), steady operation of biaxial system
is guaranteed when kc adds. It means RACCC can improve
synchronous performance and stability simultaneously.
5.2.2 Performance of RACCC with different reference model
The performance of RACCC strategy is influenced by ref-
erence model Gm(s). If bandwidth f is determined, reference
models with different damping coefficients  can be de-
signed applying the constraint conditions in section 4. Fig-
ure 10 shows the Bode plots of different reference models
with f=100 rad/s, =0.5, 1 and 1.5.
As Gm(s) is a positive real function,  cannot be less than
0.5. It can be seen from amplitude-frequency curve that
there is resonance in system when =0.5, which may deteri-
orate the stability. When =1.5, the bandwidth narrows
down which may decrease the response speed.
Figure 11 shows the experimental results when  is 0.5, 1,
and 1.5, respectively. The input of biaxial system is a ramp
trajectory. The feed rate is 125 mm/s, 0 is 45°.
Performance of RACCC strategy with different reference
models are compared in Figure 11. Comparing Figure 11(a)
with Figure 11(b), we can see that an unstable segment ap-
pears and c, e1, e2 are all bigger when =0.5. The reason is
that resonance is bad for the system stability. As  increases
from 1 to 1.5, c remains unchanged, but e1 and e2 increases.
It is because the shorter bandwidth decreases the response
speed and enlarges the tracking errors. Considering these
factors above, ξ should be chosen around 1.
6 Conclusion
In biaxial CCC motion system, model differences between
X axis and Y axis, model errors and uncertain disturbances
Figure 9 Robust adaptive cross coupling control. (a) Position trajectory;
(b) synchronous error; (c) position of X axis; (d) position of Y axis; (e)
tracking errors.
will decrease the synchronous performance and deteriorate
system stability. By analyzing the model and characteristics
of CCC system and studying effects of the factors above,
RACCC strategy is proposed. To suppress the model dif-
ferences, X axis and Y axis are forced to track the same ref-
erence model. A robust parameters adaptive law is proposed
to suppress model errors and uncertain disturbances and
make sure that individual axis can track reference model
accurately. The stability analysis of RACCC strategy is

9. 688 Chen W, et al. Sci China Tech Sci April (2016) Vol.59 No.4
Figure 10 Bode plot of reference model.
Figure 11 Experimental results of different damping factors. (a) =0.5;
(b) =1; (c) =1.5.
conducted applying Lyapunov stability theory. Experi-
mental results prove that compared with CCC strategy, the
RACCC strategy can improve synchronous performance
and stability simultaneously. In addition, if a suitable syn-
chronous error model and transfer matrix L can be defined,
the RACCC strategy is suitable to triaxial system and mul-
tiaxial system.
This work was supported by the National Basic Research Program of Chi-
na (“973” Project) (Grant No. 2013CB035600), and the National Natural
Science Foundation of China (Grant No. 51377121).
Supporting Information
The supporting information is available online at tech.scichina.com and
www.springerlink.com. The supporting materials are published as submit-
ted, without typesetting or editing. The responsibility for scientific accura-
cy and content remains entirely with the authors.
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