DOI: 10.5545/sv-jme.2024.1121 67
© The Authors. CC BY 4.0 Int. Licensee: SV-JME  Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 3-4   ▪  Y 2025
Design and Evaluation  
of a Passive Compliance Control Method  
of an Offshore Wind Turbine Blade Grinding Robot
Xinrong Liu  – Hao Li – Yu Fang – Diqing Fan
Shanghai University of Engineering Science, School of Mechanical and Automotive Engineering, China
 lxr4849@126.com
Abstract  Robots that repair offshore wind turbine blades are susceptible to interference from different factors such as external wind, which can lead to 
damage to the blades by the robot during the grinding process. Therefore, the robot needs to keep the grinding contact force constant in the complex operating 
environment. In this study, a constant force control device that is based on a pneumatic system is designed to address this problem, and a controller that is 
based on an improved Active Disturbance Rejection Control (ADRC) algorithm was proposed to control this device. Based on the analysis of the mechanism of the 
constant force control device and according to the relative order of the system, a second-order ADRC is designed. The controller utilizes a tracking differentiator 
(TD) to filter the input signal, an extended state observer (ESO) to estimate the total perturbation in the system, and a nonlinear state error feedback control law 
(NLSEF) for compensation. In order to solve the problems of electric proportional valve dead-zone characteristics, unknown interference during high altitude 
operation, tilt angle changes during grinding, dead-zone compensation, and gravity compensation algorithms were incorporated into the controller. Finally, 
the experimental platform is built to carry out experiments under various working conditions. The experimental results show that the controller improves the 
system regulation time by 59%, with an overshoot close to zero, when compared with the traditional proportional-integral-derivative (PID) algorithm. Also, both 
the absolute value of the maximum error and the mean square value of the error have been reduced to a large extent. As a result, the controller has a better 
force control accuracy and dynamic tracking performance, strong interference rejection capability and adaptability, and provides a theoretical basis for practical 
engineering applications.
Keywords  improved active disturbance rejection control, gravity compensation, dead-zone compensation, offshore wind turbine blade, pneumatic loading 
system
Highlights
 ▪ Design and mathematical model of a pneumatic constant force control device are presented.
 ▪ A controller based on an improved ADRC algorithm is proposed. 
 ▪ Dead-band compensation and gravity compensation algorithms are added to the controller.
 ▪ The improved ADRC has better responsiveness, control accuracy, and immunity to interference.
1 INTRODUCTION
With the massive consumption of fossil fuels, the problems of 
environmental degradation, climate warming, and the energy crisis 
have become more serious [1]. As well, offshore wind energy, as 
one of the very promising renewable and clean energy sources, 
can alleviate the energy pressure to a great extent [2,3]. The global 
offshore wind capacity has grown rapidly over the last decade [4]. 
China’s installed offshore wind capacity was 35.6 GW by the end 
of 2023. The increase in installed offshore wind capacity means that 
some of the installed wind turbines are ageing and so maintenance 
of the wind turbine equipment is very important [5]. The first 
maintenance of the blades should be carried out after the offshore 
wind turbine has been dynamically commissioned and has been in 
normal operation for 7 to 10 days, and then every 6 months. Offshore 
wind turbine blades, are key components of the wind turbine system, 
and are at the heart of wind turbine equipment repair [6], with repair 
costs accounting for 20 % to 25 % of the total average cost [7]. This 
includes losses due to downtime for maintenance and labor costs. 
Currently, offshore wind turbine blade surface repair primarily relies 
on manual labor with the following problems: first, the high-risk 
factor of overhead operation, high labor intensity, a grinding process 
where there is great noise and a significant amount of dust, and 
long-term work in this environment tends to lead to earlier suffering 
from occupational diseases. Second, the blade grinding quality is not 
stable, with an artificial grinding blade that has uneven quality. Third, 
blade maintenance incurs higher costs, and fourth, low efficiency. In 
order to solve the problems of manual repair, the use of robots instead 
of manually repairing the wind turbine blades is a good solution [8,9]. 
In comparison to manual repair of an offshore wind turbine blade, 
robotic repair has the advantages of high safety and efficiency [10].
If a robot is utilized in the blade polishing process, where the end 
of the polishing tool and blade contact generates the contact force, 
that contact force is too small and leads to a blade polishing quality 
that cannot meet the requirements. It should be noted that if the 
contact force is too large, an overcutting phenomenon damage to the 
surface of the blade will appear. Therefore, the size of the sanding 
force directly determines the quality of the processed workpiece, 
and how to maintain a constant sanding contact force has become a 
key issue to be solved by robot grinding. Current robotic constant 
force control methods include active compliance control and passive 
compliance control [11].
The essence of active compliance control is that the robot feeds 
back on the contact force signal and adopts a corresponding active 
control strategy to keep the contact force constant by adjusting the 
position, speed, and acceleration of the robot end, thus realizing the 
force compliance control of the grinding process. Guo et al. [12] 

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and Lakshminarayanan et al. [13] propose an adaptive variable, 
impedance active, compliance constant force control method, and an 
iterative learning controller based on impedance control. Zhang et al. 
[14] proposed a force-position anti-disturbance control strategy based 
on fuzzy proportional-integral-derivative (PID) control. Zhang et al. 
[15] proposed a constant force control based on active disturbance 
rejection control theory. However, when encountering unknown 
wind disturbances in blade grinding for offshore wind turbines, it 
is difficult to maintain the control accuracy regardless of whether 
impedance control or force-position hybrid control is adopted.
In order to cope with the problem of external disturbances, the 
researchers developed a force control end-effector [16-24], which 
achieves contact force control by means of passive compliance 
control. Wei and Xu [16] proposed the always symmetric structure 
to avoid the vibration caused by an eccentric force. Zhang et al. [17] 
utilized the second bending mode of fixed-guided compliant beams to 
design the compliance constant force mechanism. Chen et al. [18] and 
Xu et al. [19] designed a magnetic vibration absorber and integrated 
them into a smart end-effector for force control. Li et al. [20] and 
Mohammad et al. [21] use servo motors and a voice coil actuator, 
respectively, to design the end-effector to realize the control of the 
grinding contact force, but the force control is difficult. Liang et al. 
[22] and Huang et al. [23] used a low friction air cylinder to design the 
end-effector and used a PID algorithm to control it. Even though this 
compliance is better, the control accuracy is lower.
In order to solve the force control accuracy to improve the 
disturbance rejection capability, this study developed an end-effector 
using a low-friction cylinder. Also, an improved ADRC algorithm 
combining dead-zone compensation algorithm and a gravity 
compensation algorithm is proposed as the controller of this device, 
according to the principle of ADRC [24-26]. The algorithm not only 
solves the problems of the dead zone characteristics of the electric 
proportional valve and the change of the tilt angle during the grinding 
process, but also effectively copes  with the problem of the unknown 
wind interference during high-altitude operation. Finally, the authors 
conclude that through experimental verification the adjustment of 
system time is less, the maximum deviation is lower, and the tracking 
error is smaller under the improved ADRC algorithm. Therefore, it 
can be surmised that compared to the PID algorithm, the improved 
ADRC has better dynamic response performance, force control 
accuracy, and anti-interference ability.
2 METHODS AND MATERIALS
The robot is lifted by the lifting mechanism to the vicinity of the 
blade, then guided by a traction rope where the robot adheres to the 
blade by a bottom vacuum suction cup, and after that it can begin 
the repair work on the blade. Fig. 1 shows the schematic diagram of 
robotic grinding, where the robot grinds the area to be repaired by 
guiding the end-effector.
2.1 Composition and Working Theory of an End-effector
The schematic diagram of the end-effector is shown in Fig. 2, and its 
main parts are: an air compressor, a pneumatic filter and regulator 
(F.R.), a 5-way 2-position pneumatic solenoid valve, an electrical 
proportional valve, a pneumatic cylinder, an inclination sensor, a 
force sensor, a STM32, a host computer, etc. The end-effector collects 
the grinding contact force through the force sensor, and the improved 
ADRC algorithm controls the opening of the electric proportional 
valve according to the pressure feedback value so as to achieve the 
expansion and contraction of the cylinder, and to ultimately achieve 
control of the grinding contact force.
Fig. 1.  Schematic Diagram of Robot Grinding
2.2 Modelling of the End-effector
From the composition and working theory of the end-effector, it can 
be seen that the end-effector is mainly composed of an electrical 
proportional valve, a cylinder, connecting air tubes, and force 
feedback in four parts. These four parts need to be modelled next.
2.2.1 Modelling of the Electrical Proportional Valve
The flow rate through the valve port of an electrical proportional 
valve is related to the flow area of the valve port and the magnitude 
of the inlet and outlet pressures. Approximating the gas flow at the 
valve port as a one-dimensional isentropic flow of an ideal gas, with 
the SANVILLE gas flow formula, the following formula is derived:
qC Ap
RT
P
p
Cw xp
RT
P
p
qq qv













 0
1
0
0
1
0
22
 . (1)
Fig. 2.  Schematic diagram of the end-effector

P
p
k
k
k
k
k
P
P
P
P
k
k
1
0
11
1
0
2
1
0
2
11
1

























 /( )
/
,
 






















 () /
,
,
kk
t
t
P
P
C
C
P
P
1
1
0
1
0
0
1
 (2)

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C
k
t
k
k









2
1
1
, (3)
q is the mass flow rates into the electrical proportional valve;  
C
q
 is the flow coefficient; A
q
 is the orifice open area of the electrical 
proportional valve; w is the electrical proportional valve opening area 
gradient; x
v
 is the electrical proportional valve spool displacement;  
p
0 
 is the electrical proportional valve inlet pressure; p
1
 is the electrical 
proportional valve outlet pressure; R is the universal gas constant;  
T is the absolute temperature; C
t
 is the critical pressure ratio; k is the 
isentropic constant; and k is taken to be 1.4, [27].
Electrical proportional valves are electrically driven spools to 
achieve continuous control of the flow rate. When the supply port 
pressure and the temperature inside the valve is unchanged, the 
electrical proportional valve spool displacement and the input voltage 
can be equated into a proportional link, the following formula is 
derived:
xe u
vv v
= , (4)
where e
v
 is the electrical proportional valve flow gain and u
v
 is the 
control voltage.
2.2.2  Modelling of the Cylinder
The gas flows out of the electric proportional valve and into the 
cylinder, according to the law of conservation of energy:
q
dm
dt
dV
dt
q
dm
dt
dV
dt
ma
aa
mb
bb









()
,
()
,


 (5)
where q
ma
 is the inflow cylinder gas mass flow rate; q
mb
 is the outflow 
cylinder gas mass flow rate; m
a
 is the inflow cylinder gas mass; m
b
 is 
the outflow cylinder gas mass; V
a
 is the rod end volume; and V
b
 is the 
rear end volume.
Assume that the gas is an ideal gas and satisfies the formula:
PR T
d
 , (6)
where P
d
 is the gas pressure in the cylinder; ρ is the gas density in the 
cylinder, and T is the gas temperature in the cylinder.
Assuming that the spool pushes the cylinder downward, the 
cylinder displacement is x; and the cylinder stroke is l. From Eq. (6), 
it follows that:
qP x
A
RT
P
lx
kRT
qP x
A
RT
P
lx
kRT
ma a
a
a
mb b
b
b


 










(/ )
(/ )
2
2
 

, (7)
where k is the adiabatic index, k = 1.4; A
a
 is the contact area of the 
without rod chamber of the cylinder; and A
b
 is the contact area of the 
with rod chamber.
2.2.3 Modelling of the Pneumatic Hose
The gas flow from the electric proportional valve passes through a 
pneumatic hose of length L and enters the chamber of the cylinder 
without a rod. The gas flow in the tube is equal to the outlet flow 
of the proportional valve. According to Anderson’s theory, Eq. (8) is 
derived:
qK PP 
11 2
() , (8)
K
DA
L
v
1
2
32



, (9)
where q is the gas mass flow rate of the g pneumatic hose; P
1
 is the 
inlet pressure of the pneumatic hose; P
2
 is the outlet pressure of the 
pneumatic hose; ρ
v
 is the average density of the system gas; D is the 
inner diameter of the gas pipe; μ is the viscosity coefficient of the 
system gas; and A is the cross-sectional area of the pneumatic hose. 
From Eqs. (1), (7) and (8), it follows that:



P
kPxA
lx
kRTCWx PH PPqK
lx
P
a
aa
qv a
b







(/ )
(/ (/ ))
(/ ) 22
00 11

k kPxA
lx
kRTCWx PH PPqK
lx
bb
qv b

(/ )
(/ (/ ))
(/ )
.
22
00 11












 (10)
2.2.4  Modelling of the Force Sensor
The force sensor belongs to the resistive type sensors, and is 
considered as an equivalent proportional element, from which Eq. 
(11) is derived:
uK F
u
= , (11)
u is the output voltage; K
u
 is the voltage gain; and F is the detection 
pressure.
Since the force transducer has a good linear relationship, it is 
linearized and the following formula is derived.
FK x
m
= , (12)
K
m
 is the pressure gain; x is the output signal.
2.2.5  End-effector Force Balance Analysis
The end-effector force sketch is shown in Fig. 3. According to 
Newton’s second theorem, without considering the effect of cylinder 
friction, and the cylinder piston force, the balance equation is:
PA PA ms Bs F
aa bb n
    , (13)
where A
a
 is the contact area of the rear end of the pneumatic cylinder 
and A
b
 is the contact area of the rod end; P
a
 is the rear end of the 
cylinder pressure; P
b
 is the rod end pressure; m is the cylinder piston 
and load total mass;  s is the cylinder piston moving speed;  s is the 
cylinder piston moving acceleration; B is the damping coefficient; F
n
 
is the end-effector grinding contact force.
Fig. 3.  Schematic diagram of the end-effector
When the robot performs grinding work, the spatial attitude 
changes in real time, and the grinding contact force collected by the 
force sensor has a large error due to the gravity effect of the end-
effector. Therefore, it is necessary to design a gravity compensator 
to eliminate the gravity influence of the end-effector along the force 
control direction. The force analysis of the end-effector is shown in 
Fig. 3, where mg is the gravity of the end-effector, F
d
 is the force 
measured by the force sensor, and θ is the angle between the gravity 
and the force measured by the force sensor. Therefore, the actual 
output force F
n
 is:
FFmg
nd
cos.  (14)

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When the end-effector is in contact with the blade, the axial force 
is F
n
, the resulting displacement is y, and the equivalent stiffness 
model coefficient is K
e
, the following formula is derived:
FK y
ne
= . (15)
From Eqs. (13-15), it follows that:
PA PA Ms Bs Ky
aa bb e
    . (16)
To facilitate the analysis and design of the controller, from Eqs. 
(1-16). If we define the state variable F = x
1
, 

Fx =
2
, P
a
 = x
3
, P
b
 = x
4
, 
x
v
 = u, then the system state space equation set is obtained as:




xx
xK Ax Ax Bs Ky
x
kPxA
lx
kRTCWx
na be
aa
q
12
23 4
3
2

 




()
(/ )
v va
bb
qv
PH PP qK
lx
x
kPxA
lx
kRTCWx PH
00 11
4
0
2
2
(/ (/ ))
(/ )
(/ )




 

0 01 1
1
2
(/ (/ ))
(/ )
,
PPqK
lx
yx
b
















 (17)
3  IMPROVED ADRC CONTROLLER DESIGN
If we take Eq. (17), the end-effector model described as a 3
rd
 order 
system, then according to the method of active disturbance rejection 
controller order selection [28], the relative order of the system is taken 
to be 2, the 3rd order is regarded as a perturbation, and the second-
order nonlinear active disturbance rejection controller is designed 
and Eq. (17) is rewritten as:
 
   
Ff FPPw tb u
ab
w


 ,,,, ,
(.)
 (18)
where w(.) is a nonlinear function of the “total disturbance” of 
the end-effector, including the unmodelled portion, higher-order 
nonlinearities, and control gain estimation errors, and b is the control 
gain estimate.
Fig. 4. End-effector controller
The improved ADRC controller consists of a tracking 
differentiator, a nonlinear state error feedback control law, 
an extended state observer, gravity compensation, dead-zone 
compensation of five parts, as shown in Fig. 4.
3.1  The Tracking Differentiator
The tracking differentiator is used to smooth the desired force and 
reduce the overshoot caused by excessive error. According to Han 
[28], the tracking differentiator (TD) is as follows:


r
rr x
r
fhan rR h
d
12
21 2

 



 ,,,
,     , (19)
where x
d
 is the desired force loading input signal; r
1
 is the tracked 
value of the input signal x
d
, and r
2
 is the differential of the input 
signal x
d
. The fhan(z
1
, z
2
, R, h) function is as follows:
dh hz yza
ad dy
aasign ya d
ss
Ra  

 

2
02 10
1
20 1
8
2
,,
(| |)
() () /
[
y
i igny ds igny d
aayasa
ss igna ds igna
() () ]/
()
[( )(
 
 
 
2
02 2 y
a
 
 













d
fhan Ra ds igna sr sign a
)]/
[/ () ]( )
,
2
0 a
 (20)
where z
1
 and z
2
 are the state quantities of the fhan; R and h are 
the control parameters of the fhan. R depends on the smoothing 
processing speed and the system capacity; the larger the value of R, 
the stronger the ability to track the input signal. The value of r is 
usually not infinite, and so it needs to be adjusted according to the 
actual situation. Additionally, h is the step size.
3.2  Extended State Observer
The extended state observer is used to estimate the total perturbation 
of the system in real time and is the central aspect of the active 
disturbance rejection control algorithm. The extended state observer 
(ESO) derives a state estimate x

1 , x

2 of the desired signal and an 
estimate of the total disturbance x

3 from the total disturbance w(.), 
output signals y and  y . The ESO design is as follows:
exy
xx e
xx fale hb u
xf al
0
1
12
10
23
20 1
3
3


 











(,,)
( (, ,)
,
eh
02










 (21)
β
1
, β
2
, β
3
, α
1
, α
2
, h is the parameter of ESO and the fal() expression 
is as follows:
fale h
eh eh
sign ee eh
(,,)
/,
() ,
,
()
0
0
1
0
00 0












 (22)
3.3  The Nonlinear State Error Feedback Control Law
The nonlinear state error feedback (NLSEF) is a nonlinear 
control combination used instead of the linear combination of the 
conventional PID controller. Combining the observations of ESO 
with the output of TD, the combination NLSEF is as follows:
erx
erx
ukfale hkfale h
pd
11
1
22
2
01 12 2


 








(, ,) (, ,) 
 
, (23)
where k
p
 and k
d
 are the NLSEF nonlinear combination coefficients, 
respectively. In order to reduce the model uncertainty and the effect 
of external perturbations, x

3 is compensated to the control quantity 
with the compensation equation:
u
ux
b


0
3

. (24)
3.4  The Dead-zone Compensator
The dead-zone compensation is an algorithm to make the spool of 
the electrical proportional valve jump out of the neutral dead-zone 
quickly and enter the control zone. The dead-zone model parameters 

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are obtained by experimental identification methods to achieve 
pre-compensation for a wide range of dead-zones. The dead band 
compensator is designed as follows:
u
uu
uu
u
u
f








3
4
0
0
,
,
, (25)
where u
f
 dead-zone compensation after the control amount; u
3
 for the 
experimentally measured electrical proportional valve inlet minimum 
operating voltage; u
4
 for the experimentally measured electrical 
proportional valve maximum operating voltage.
4  RESULTS AND DISCUSSION
When repairing offshore wind turbine blades, the external 
environment is very complex, and at the same time there will be wind 
and other disturbances in the high-altitude air to increase the difficulty 
of the whole grinding operation. The robot controls the end position 
in the sanding process while using the end-effector to correct the end 
contact force, so as to realize the decomposition of force control and 
position control in the grinding process. This makes it possible to 
complete offshore wind turbine blade grinding even when the entire 
grinding process is disturbed by the external environment. In order 
to verify the application of the improved ADRC algorithm in the real 
working environment, an experimental rig was built and simulated 
offshore wind turbine blade grinding experiments were carried out. 
The robot model is Elibot ec66 with a working radius of 914 mm 
and a repeat positioning accuracy of ±0.02 mm. Both algorithms use 
image processing and vision-guided path-planning-based methods. 
The test bench is shown in Fig. 5, and the main components and 
parameters of the experiment are shown in Table 1.
Fig. 5.  Laboratory bench
Table 1.  Key Components and Parameters
Component Parameters
Air source Outlet pressure: 1.0 MPa
F. R. Set pressure: 0.9 MPa
Electrical proportional valve Max. pressure: 0.9 MPa; Sensitivity: 0.2 %
Pneumatic cylinder Stroke: 20 mm
Inclination sensor Angular range: ±90°, Repeatability: 0.5°
Force sensor Capacity: 0 N - 100 N; Sensitivity: 0.1 % 
4.1  Dead-zone Compensation Experiment
Firstly, it is verified whether the improved ADRC algorithm is able 
to compensate for the dead-zone effectively. Since the effect of the 
dead-zone is mainly manifested during commutation, a sinusoidal 
signal is used as the input signal. In the formula, where the input 
signal is F = 60 + 10 sin(1.257t), the amplitude is 20 N, and the force 
tracking curve and force tracking error curve are shown in Fig. 6. 
As can be seen in Fig. 6, the input signal can be tracked well with 
the dead-zone compensation algorithm; whereas without the dead 
zone compensation algorithm there is a hysteresis that leads to a 
large error. Therefore, this algorithm enables the controller to cross 
the dead-zone quickly and improves the dynamic performance of the 
system.
Fig. 6. Dead-zone compensation experiment
4.2  Gravity Compensation Experiment
When the end-effector is tilted, the grinding contact force is varied 
by the gravity of the device. This experiment, in order to verify the 
compensation effect of the gravity compensation algorithm, sets the 
desired force to 60 N, the grinding inclination angle is 0º to 90º, and 
the output force change curve is shown in Fig. 7. Without the gravity 
compensation algorithm, the output force error increases with the 
inclination angle, reaching a maximum of 18.7 N, while with the 
gravity compensation algorithm, the output force is smooth, and the 
maximum error is 2.3 N.
Fig. 7.  Gravity compensation experiment
4.3  Constant Force Loading Experiment
The constant force loading experiment can test the transient response 
performance of the system. The desired force is set to F = 60 N 
according to the real grinding situation. The output force response 
curve and force tracking error curve are shown in Fig. 8. The system 
regulation time of the PID control is 1.83 s, with a maximum 

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overshoot of 10.93 %; while the system regulation time of the 
improved ADRC control is 0.75 s, with an overshoot close to zero. 
Therefore, the improved ADRC exhibits better transient response 
performance than PID control in the control system.
Fig. 8.  Constant force loading experiment
4.4  Sinusoidal Force Loading Experiment
Sinusoidal loading experiments can test the dynamic performance of 
the whole system. The desired force is set to F = 60 + 10 sin(1.257t) in 
the experiment. The output force response curve and force tracking 
error curve are shown in Fig. 9. The maximum force tracking error is 
5.2 N for PID control and 1.3 N for improved ADRC control. It can 
be obtained that the improved ADRC algorithm exhibits a smaller 
force tracking error, a smoother trend, and higher quality of control 
than PID control.
Fig. 9.  Sinusoidal force loading experiment
4.5  Triangular Force Loading Experiment
Triangular force loading experiments can test the responsiveness of 
the system to sudden changes in the input signal. The triangular force 
signal has a period of 5, an amplitude of 20, and an initial value of 
50. The output force response curve and force tracking error curve 
are shown in Fig. 10. It can be seen in the output force response 
curve and force tracking error curve that the PID control has some 
hysteresis when the force signal changes suddenly. Compared to the 
PID algorithm, the improved ADRC algorithm has a faster response 
and less force tracking error.
4.6  Trapezoidal Force Loading Experiment
Trapezoidal force loading can further test the fast performance and 
sudden change response performance of the controller, and the output 
force response curve and force tracking error curve are shown in 
Fig. 11. When the direction of the loading signal changes abruptly, 
the control effect of the improved ADRC algorithm is significantly 
better than that of the PID algorithm, showing better anti-interference 
ability and stronger robustness.
Fig. 10. Triangular force loading experiment
Fig. 11.  Trapezoidal force loading experiment
4.7  Constant Force Loading Interference Experiment
In order to further verify the anti-disturbance performance of the 
improved ADRC algorithm, this experiment simulates the effects of 
the unknown wind disturbance and the deformation of the suction 
cup of the robot base at the high altitude. In the constant force 
loading process of two different algorithms, two different directions 
of disturbance forces are added. The output force response curves and 
force tracking error curves are shown in Fig. 12. From Fig. 12, it can 
be concluded that the improved ADRC algorithm is much stronger 
than the PID algorithm in the case of adding the same disturbance.
Fig. 12.  Constant force loading interference experiment

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4.8  Constant Force Loading Interference Experiment
In order to better quantitatively compare the two control algorithms, 
two sets of data, the absolute value of the maximum error M
e
 and 
the mean-square value of the error μ
e
, are introduced for further 
illustration:
Me i
e
iN
 

max( ),
,, ..., 12
 (26)

e
i
N
N
ei  


1 2
1
() . (27)
The results are shown in Table 2. Comparing the performance 
indicators from the Table 2, it can be seen that the absolute value 
of the maximum error and the mean square value of the error of the 
improved ADRC algorithm in several force loading experiments are 
smaller than that of the PID control. From this, the improved ADRC 
algorithm is superior to the PID control algorithm in terms of system 
regulation time, dynamic response ability, and anti-interference 
ability, which prove the effectiveness of the algorithm in the control 
of the contact force of end grinding of a high altitude wind turbine 
blade grinding robot.
5  CONCLUSIONS
In this paper, an end-effector based on passive flexible control 
is designed for the problems of unknown wind interference and 
deformation of suction cups at the robot base when the robot is 
grinding high-altitude wind turbine blades. The device is fixed at 
one end of the robot end while the other end fixes the grinding tool, 
effectively controlling the grinding contact force.
The end-effector adopts a pneumatic control system. Several 
problems such as electrical proportional valve dead-zone, attitude 
transformation, and external interference appear. In this paper, by 
establishing a mathematical model of the control system and analyzing 
the influence of system parameters and dead-zone characteristics on 
force control accuracy, an improved third-order ADRC controller 
with dead-zone compensation and gravity compensation is designed. 
It not only solves the problems of system lag caused by dead zone and 
attitude transformation affecting the contact force, but also improves 
the anti-interference ability of the control system.
Finally, seven sets of experiments are conducted to make a 
comprehensive comparison of the control algorithms. It can be 
concluded that the improved ADRC algorithm is more stable, faster, 
and more accurate than the PID algorithm in terms of the robustness 
of the control system and the anti-interference ability.
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Table 2.  Performance indices
Control method Performance indices Constant loading Sinusoidal loading Triangular loading Trapezoidal loading Constant loading interference
PID
M
e
6.56 4.89 4.29 20.73 5.74
μ
e
4.27 1.32 0.92 3.05 2.14
ADRC
M
e
0.52 1.24 1.14 20.16 1.89
μ
e
0.12 0.13 0.13 2.59 0.26

Mechatronics
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Acknowledgements The research work in this paper was supported by a 
grant from the Institute of Automation Equipment and Application, Shanghai 
University of Engineering Science.
Received 2024-08-06, revised 2024-11-12, 2025-01-08, accepted 2025-
01-14, Original Scientific Paper.
Data availability The data that underpin the results of this research are 
available for use by other researchers. Additionally, the corresponding author 
can provide additional raw data upon reasonable request. 
Author contribution Xinrong Liu: supervision, project administration, 
methodology, writing – review & editing; Hao Li: methodology, formal analysis, 
validation, writing – original draft; Yu Fang: conceptualization, methodologa, 
writing – review & editing. Diqing Fan: formal analysis, data curation. All 
authors have contributed significantly to this work and thoroughly reviewed 
and approved the final version of the manuscript.
Zasnova in vrednotenje pasivne metode nadzora skladnosti robota 
za brušenje lopatic vetrnih turbin na morju
Povzetek Roboti, ki popravljajo lopatice vetrnih turbin na morju, so 
dovzetni za motnje različnih dejavnikov, kot je zunanji veter, zaradi česar 
lahko robot med postopkom brušenja poškoduje lopatice. Zato mora robot v 
neugodnem delovnem okolju ohranjati konstantno kontaktno silo brušenja. 
V tej študiji je bila za reševanje tega problema zasnovana naprava za nadzor 
konstantne sile, ki temelji na pnevmatskem sistemu. Za nadzor te naprave 
pa je bil predlagan krmilnik, ki temelji na izboljšanem algoritmu ADRC (Active 
Disturbance Rejection Control). Na podlagi analize mehanizma naprave 
za nadzor konstantne sile in glede na relativni red sistema je bil zasnovan 
ADRC algoritem drugega reda. Krmilnik uporablja sledilni diferencial (TD) 
za filtriranje vhodnega signala, razširjeni opazovalnik stanja (ESO) za oceno 
celotne motnje v sistemu in nelinearni zakon povratne kontrole napake 
stanja (NLSEF) za kompenzacijo. Da bi rešili težave z značilnostmi mrtve cone 
električnega proporcionalnega ventila, neznanimi motnjami med delovanjem 
na veliki nadmorski višini, spremembami kota nagiba med brušenjem, so bili 
v krmilnik vključeni algoritmi za kompenzacijo mrtve cone in kompenzacijo 
težnosti. Na koncu je bila zgrajena eksperimentalna platforma za izvajanje 
poskusov v različnih delovnih pogojih. Rezultati poskusov kažejo, da krmilnik 
v primerjavi s tradicionalnim proporcionalno-integralno-derivativnim (PID) 
algoritmom izboljša čas regulacije sistema za 59 %, pri čemer je prekoračitev 
blizu nič. V veliki meri sta se zmanjšali tudi absolutna vrednost največje 
napake in srednja kvadratna vrednost napake. Posledično ima krmilnik 
boljšo natančnost krmiljenja sile in zmogljivost dinamičnega sledenja, močno 
sposobnost zavračanja motenj in prilagodljivost ter zagotavlja teoretično 
podlago za praktično inženirsko uporabo.
Ključne besede izboljšani aktivni nadzor zavrnitve motenj, gravitacijska 
kompenzacija, kompenzacija mrtvega območja, lopatica vetrne turbine na 
morju, pnevmatski obremenilni sistem