Proceedings VCM 2012 108 mô hình maxwell slip để bù ma sát trong điều khiển khớp mềm khí nén

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Mô hình Maxwell-Slip để bù ma sát trong điều khiển khớp mềm khí nén
một bậc tự do: một bộ bù Lead-Lag phi tuyến tương đương
Maxwell-Slip Model for Hysteresis Compensation
in Controlling an 1-DOF Pneumatic Soft Joint:
An Equivalently Nonlinear Lead-Lag Compensator
Võ Minh Trí
Trường Đại học Cần Thơ
e-Mail:
Tóm tắt
Một trong những ứng dụng điển hình của cơ nhân tạo khí nén là khớp đôi, được tạo thành từ một cặp cơ.
Điều khiển khớp này vẫn còn còn nhiều vấn đề cần nghiên cứu, bởi (i) tính phi tuyến của hệ thống khí nén, (ii)
trễ phi tuyến xảy ra trong lớp vỏ của cơ bắp, và (iii) tính biến đổi theo thời gian. Trong hầu hết các nghiên cứu
gần đây, mô hình khớp này thường được tính gần đúng, tính phi tuyến và tính bất định (nếu có) của mô hình đã
được khắc phục bằng hệ thống điều khiển đi kèm. Trong nghiên cứu này, mô hình khớp đã được phát triển bao
gồm cả phần trễ của áp suất theo vị trí, theo đó phần này được mô tả chính xác bằng cách sử dụng mô hình
Maxwell-Slip (MS). Mô hình này rất thích hợp để nhúng vào các chương trình điều khiển để bù sự trễ. Hệ
thống điều khiển đi kèm do đó đơn giản hóa và chỉ cần áp dụng một bộ điều khiển PID thông thường. Hoạt
đồng của hệ thống điều khiển vị trí của khớp quay với đáp ứng bước rất hiệu quả khi bộ bù trễ hoạt động. Phân
tích FRF cũng cho thấy rằng việc bù trực tuyến tạo ra một kiểu giống với bộ bù phi tuyến lead-lag.
Abstract
One of the typical applications of pneumatic artificial muscles (PAMs) is the antagonistic manipulator,
which is made up of a pair of PAMs. Control of this manipulator is still an on-going issue, facing with (i) the
nonlinearity of a pneumatic system, (ii) the nonlinear hysteresis occurring in the PAM sheath, and (iii) the
time-varying problem due to the PAM creep. In most recent studies, the manipulator model is a rough
approximation, and the nonlinearities and/or the model uncertainties were left to be overcome by the
associated control system. In this paper, the novel joint model was developed including the extracted pressure
difference-joint angle hysteresis part, which can be accurately described by using the Maxwell-Slip (MS)
model. This hysteresis model is suitable for embedding in the control scheme in terms of hysteresis
compensation. The associated control system is therefore simplified to the application of a conventional PID

controller. The position control performance of the manipulator with step response showed effectively
improved when the hysteresis compensation is introduced. The FRF analysis also showed that the online
compensation generated a kind of equivalently nonlinear lead-lag compensator.

Nomenclature
DOF Degree of Freedom
PAM Pneumatic Artificial Muscle
MS
PID
FRF
HC
Maxwell-Slip
Proportional Integral Derivative
Frequency Response Function
Hysteresis Compensation

1. Introduction
One of the typical applications of a pneumatic
artificial muscle (PAM) is the antagonistic
manipulator, which is made up of a pair of
antagonistically arranged PAMs, simply called
muscle. This one degree-of-freedom pneumatic
muscle manipulator can operate as a soft joint
owing to its inherent compliance with a well-
known advantage, i. e. its angular position and its
stiffness can be controlled independently. This
setup is considered as a common case-study for
most researchers before they go to more complex
and multi-degree-of-freedom robot manipulators
[1, 2, 3, 4, 5, 6], or for those who want to

investigate the control problems involved [7]. The
antagonistic system manifests the advantages of
the individual muscles, such as large
power/volume or power/weight ratios, compliance,
direct power, cleanliness, low cost, etc. However,
the existing drawbacks of the individual muscles
come also to exist in the coupled system, i.e. poor
performance in position control due to the load
variations affecting on the position [2], and the
796 Võ Minh Trí
VCM2012
inherent hysteresis and muscle creep [8] which
leads to nonlinearities and uncertainties in the
system.

The occurrence of hysteresis, muscle creep and
model uncertainties appear to be the most
challenging for controlling the pneumatic muscle
manipulator. As difficult as finding an accurate
model for an individual PAM, the investigators
usually approximate the manipulator dynamics
with a rough model. Also muscle creep due to the
environmental effects on the muscle sheath, results
in extra degrees of model uncertainties. In
addition, the hystereses of the two PAMs do not
cancel each other when integrated in the
antagonistic system, leading to nonlinearities in
this coupled system. These nonlinearities and
uncertainties are a source of difficulty for
developing a position control which is still an on-

going issue challenging researchers. Literature
shows that most investigators choose for
developing advanced control algorithms to control
the joint position. Some advanced techniques were
proposed to overcome the nonlinearities and
uncertainties such as pole placement methods [1],
neural networks control [8,14], variable structure
control or sliding mode control [15,16], or using a
combination of more advanced techniques such as
nonlinear PID control [6], adaptive fuzzy logic
sliding mode control [17]. Most of these studies
investigate on tracking position problems. A few
authors [11, 15] consider the load effects.

In this work, all possibilities leading to the
nonlinearities, uncertainties, and time-varying
parameters are carefully analyzed and synthesized
in an empirical model. In this approach, the
constraint model of the static torque of the
antagonistic manipulator joint, incorporated with
the muscle creep, is determined [18]. The
separated hysteresis using the constraint model
makes it nullified to the creep effect. This
hysteresis is in the form of the extracted pressure
difference/joint angle hysteresis which is found
suitable to be implemented in the control scheme
in terms of hysteresis compensation as shown in
Fig.1.

The paper is organized as follows. Section 2

describes the experiment apparatus and defines the
system variables. Section 3 shows how to model
the pressure difference/angle hysteresis. Section 4
presents the experimental results supported by an
extensive discussion. Section 5 closes with the
conclusions and future work.

2. Experimental setup
The test setup consists of a pair of FESTO fluidic
muscles (type MAS-20-200N) configured into an
antagonistic configuration. A pair of FESTO
directional proportional valves (type MYPE-5-
M5-010B) was used to exactly control the pressure
in each muscle. Two similar pressure sensors
(SENSORTECHNICS type of PTE5151D1A)
were mounted between the valves’ outlet and the
muscle inlet to provide feedback for the pressure
control loop. In order to create a rotary joint, one
muscle end was connected to the other via a timing
belt that spanned a timing gear, which was
mounted on an axis with two supporting bearings.
Two coupling adaptors were designed to connect
the other ends of the two muscles to the load cells
(type DBBP-200 from BONGSHIN), which were
fixed to the support. The joint axis and the load
cell support were fixed to a rigid frame. The joint
rotation was measured by an incremental rotary
encoder (PANASONIC type of E6C-CWZ1C-M).
An arm or a lever, fixed to the joint axis, was used
to exert an external torque to the joint. All I/O

information from/to the plant setup was processed
by a 16-bit data acquisition card DAQmx NI-6229
from National Instruments, embedded in a real-
time desktop PC. The control and measurement
algorithms were developed based on LabView
Professional Development System for Windows
with the add-on LabView Real-Time Module.

In the two branches of the setup (e.g. agonistic and
antagonistic) the components and control elements
(Fig. 2) were selected identically such that the
behaviour of the antagonistic system is symmetric
with respect to the centre (initial) position. The
initial position is also called the zero position since
with our design the joint angle starts at 0 degrees
at this position.


PI
Muscle volume
-
+
d


r




r
P


d
P




Mechanical system
Hys. Compensation
+
+
hys
P


H1. Proposed control scheme of antagonistic setup with
hysteresis compensation
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H. 2 The antagonistic setup

The joint angle

starting from the centre position
is positive, measured clockwise and negative,
measured counterclockwise.

The pressure difference,
1 2
p p p
  
, plays a role
as input to activate the joint motion, which is
considered as the output of the control system.
1 2
( )
T r F F
  is the joint torque
Subscript 1, 2 indicate the right-hand side muscle
and the left-hand side muscle
The constraint model takes the form:
( ) ( )
const
T a p b
 
   (1)
where
( )
a

is the constraint model’s slope and
( )
b

the constraint model’s intercept. They are
functions of the joint angle [18].
The model of Eq. 1 allows us to calculate the

pressure difference needed to act against the
torque exerted at a certain joint angle as follows:

( )
( )
T b
p
a



  (2)

3. Modeling
3.1 Pressure difference/joint angle hysteresis
extraction
In the isotonic condition test [13], the pressure
differences needed to move the external payload
forward and backward, are different. This is not
due to the gravity torque acting on the joint,
which, depends only on the joint position, not on
the direction of motion. Meanwhile, the force
generated by each muscle depends on the motion
direction due to hysteresis effects [12]. If the
required torque generated by the coupled muscles
to counter the gravity torque which is regardless of
the motion direction, the pressure of each muscle
has to adapt automatically to the requested force
that is contributing to the joint torque. As a result,
the pressure differences needed at the same

equilibrium position of the pendulum arm will be
different depending on which direction the joint is
moving towards. The pressure difference
hysteresis loops linked to the joint motion, are
shown in Fig. 3.
-40 -30 -20 -10 0 10 20 30 40
-5
-4
-3
-2
-1
0
1
2
3
4
5
Joint angle (degree)
Pressure difference (bar)


load 1kg
load 2.75kg
load 4.5kg
simulated at 1kg

H. 3 The isotonic measurements at different
loads compared to simulated pressure
difference reconstructed from the constraint
model.

This hysteresis is called the pressure
difference/joint angle hysteresis. It behaves
similarly to the torque/joint angle hysteresis
exposing nonlocal memory and quasi rate
independency. This type of hysteresis can be
modeled by using the Maxwell-slip model [18].
From this figure, it can be observed that the
heavier the load, the shorter the interval motion of
the pendulum arm and the higher the slope of the
hysteresis loop will be. If each pressure
difference/joint angle hysteresis at an isotonic
condition would be modeled separately. This
would lead to a number of models which can be
avoided as follows. As discussed earlier, the
constraint model was formed in the condition
without hysteresis. From this model, the pressure
difference needed for a certain exerted torque can
be predicted exactly using (Eq.2). This pressure
difference is also called the constraint pressure
difference. However the pressure difference/joint
angle hysteresis is inevitable and makes the actual
pressure difference needed a bit higher than the
constraint pressure. The actual pressure difference
needed is referred to as the isotonic pressure
difference and is governed by Eq 3. If a
comparison is made between the measured (loop
2
c
U
2

P

s
P
2
CV
s
P
1
CV
1
P
1
c
U




0


P: muscle pressure



θ: joint angle

P
s

: supply pressure r: timing gear
radius
CV: control valve F: muscle force
U
c
: valve control signal
Δ
P=P
1
-
P
2
: pressure difference

2
F
1
F
r
T

T

1
Muscle
2
Muscle
arm
payloa
d

798 Võ Minh Trí
VCM2012
type) and the simulated (single line) at the same
load (for example a 1 kg load as shown in Fig. 3),
the extracted hysteresis loop of a heavier load is
always enclosed by the extracted hysteresis loop of
a lighter load, as shown in Fig. 4. From this figure,
it may be visibly concluded that all of them match
the shape of hysteresis with nonlocal memory
[12]. This allows us to select the biggest loop to
derive the model. This loop is referred to as a
global loop since it can represent all other smaller
local loops corresponding to the smaller loads.
isot cons hys
p p p
     (3)
hys isot cons
p p p
     (4)
where
isot
p
 is the actual pressure difference
measured from the isotonic tests,
cons
p
 is the
pressure difference reconstructed from the
constraint model with respect to the measured

torque and joint angle in the isotonic tests, and
hys
p
 is the extracted hysteresis resulting from
differentiating between the actual pressure
difference and the reconstructed pressure
difference.

3.2 Modeling process
The procedure to model hysteresis using the MS
model has been discussed in previous papers [12,
13, 18] and is applied in this paper. The boundary
of the global hysteresis loop is geometrically
defined as the dotted curve also shown in Fig. 4.
Notice that the intuitive selection of the
asymptotes for identifying the MS elements is
usually made by visual inspection of the of the
hysteresis loop, in which the trade-off between the
model accuracy and the nonlinear behavior of the
two muscle interaction close to the outmost
pendulum positions is considered. Four steps of
hysteresis identification are recalled hereby:
 Obtain the hysteresis loop experimentally.
 Shrink the upper (or lower) half of hysteresis
loop to get the virgin curve.
 Pick up intuitively the segments which are
asymptotes to the virgin curve.
 Calculate the stiffness and saturation level for
each element by establishing and solving a
system of equations that is formulated based

on the coordinates of the end points of the
selected segments on the plot.
-40 -30 -20 -10 0 10 20 30 40
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Joint angle (degree)
Extracted hysteresis (bar)


load 1kg
load 2.75kg
load 4.5kg

H. 4 The extracted hysteresis loops: the bigger
loop corresponding to the lighter load
encloses the smaller loop of the heavier
load. The shape of the loops matches that of
hysteresis with nonlocal memory.

4. Results and discussion
4.1 Pressure difference/joint angle hysteresis
capture
-40 -30 -20 -10 0 10 20 30 40

-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Joint angle (degree)
Extracted hysteresis (bar)


measured
modelled

H. 5 Extracted pressure difference, captured for
arbitrary joint trajectories
The boundary of the global loop shown in Fig. 4 is
constructed by the double-stretched virgin curve
(upper half loop) and flipped-double-stretched
virgin curve (lower half loop). Based on this
boundary, the virgin curve is withdrawn, and the
parameters for four MS elements are calculated
and shown in Table I, where ‘k’ and ‘w’ are slopes
and saturation values respectively of each
Maxwell-element 1 to 4. These parameters are
used for online simulation and to compare the
measurement to the extracted hysteresis loop. Fig.
5 illustrates that the model captures very well the

experimental hysteresis loops. The mismatch of
the model could be foreseen as a tradeoff between
the model accuracy and the nonlinear regions had
to be made beforehand when picking up the MS
elements. It means that we accept larger model
errors in the outmost pendulum positions while
reducing the errors at rest of the joint motion
interval, as shown in Fig. 6. This compromise is
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based on the fact that close to the outermost
positions, the pressure loop has a higher damping
ratio owing to a friction flow (Fig. 7, at 40-degree
amplitude of excitation), while in the middle range
we try to keep a better fit between the model and
the measurements, for serving hysteresis
compensation later on.

Table 1. Four Maxwell-slip elements’ parameters
1 2 3 4
k 0.0557 0.0170 0.0021 0.0033

w 0.1680 0.1383 0.0342 0.1335


-40 -30 -20 -10 0 10 20 30 40
-0.25
-0.2
-0.15
-0.1

-0.05
0
0.05
0.1
0.15
0.2
0.25
Joint angle (degree)
Model error (bar)

H. 6 Model errors as a function of the joint
angle, increasing in the outermost positions.

4.2 Performance of the manipulator joint
through hysteresis compensation.
Figure 7 demonstrates the position control
performance of the system with and without
hysteresis compensation.
0 500 1000
-10
0
10
Amplitude 10 degree
0 500 1000
-40
-20
0
20
40
Amplitude 30 degree

0 500 1000
-20
0
20
Amplitude 20 degree
time (x20ms)
0 500 1000
-50
0
50
Amplitude 40 degree
time (x20ms)

H. 7 Position response of the antagonistic system
with 1kg payload at different excitation
amplitudes with and without hysteresis
compensation.

The system was excited under a square wave of
0.1 Hz, using 4 different amplitudes of 10, 20, 30,
and 40 degrees. In each subplot, two cycles are
illustrated; the first one is the system response
without hysteresis compensation while the second
one is the system response with hysteresis
compensation. Notice that the PI controller
parameters were tuned by trial-and-error, and kept
unchanged during the test performance. Based on
this result, we recognize that:

10

0
-60
-40
-20
0
Amplitude 3deg
[

r
/

d
] (dB)
10
0
-60
-40
-20
0
Amplitude 6deg
10
0
-60
-40
-20
0
Amplitude 9deg
10
0
-60

-40
-20
0
Amplitude 12deg
10
0
-300
-200
-100
0
Hz
Phase (deg)
10
0
-300
-200
-100
0
10
0
-300
-200
-100
0
10
0
-300
-200
-100
0

10
0
-60
-40
-20
0
Amplitude 3deg
[

p
r
/

d
] (dB)
10
0
-60
-40
-20
0
Amplitude 6deg
10
0
-60
-40
-20
0

Amplitude 9deg
10
0
-60
-40
-20
0
Amplitude 12deg
10
0
-300
-200
-100
0
Hz
Phase (deg)
10
0
-300
-200
-100
0
10
0
-300
-200
-100
0
10
0

-300
-200
-100
0

H. 8 FRF measurement of the pressure loop
responses to the different amplitudes of
position excitation: (a) without hysteresis
compensation, (b) with hysteresis
compensation.
 In general, before and after hysteresis
compensation is activated, the occurrence of
the oscillation is nonlinear with respect to the
amplitude of excitation. The system is more
oscillating at 20 degrees than at 10 degrees,
but is less oscillating at 30 and 40 degrees than
at 20 degrees. This phenomenon could be a
result of increased nonlinear flow effects at
higher pressure buildup as discussed in [13].
 The compensation action provides a certain
damping ratio to the system. The effectiveness
a)

b)

800 Võ Minh Trí
VCM2012
of the compensation indicates that it is
appropriate to reduce the oscillation around the
desired position.

4.3 FRF analysis for pressure loop
To perform the FRF measurements, we excite the
system with a multisine of 10Hz of the interest
band, and 4 different amplitudes, i.e. 3, 6, 9, and
12 degrees, to reveal the nonlinearities of the
system. To guarantee the symmetric behaviour of
the system, it was only tested around its centre
position. The pressure block reflects the
interaction between the response position and the
pressure loop since there the response position is
fed back. Both cases, with and without hysteresis
compensation, are compared.
-60
-50
-40
-30
-20
-10
0
Magnitude[

r
/

d
] (dB)


10

-1
10
0
10
1
-270
-180
-90
0
Phase (deg)
Bode Diagram
Frequency (Hz)
w ithout HC
w ith HC

H. 9 FRF measurement of the pressure loop
responses to the 12-degree amplitude of the
position excitation: (a) with/without
hysteresis compensation, (b) with
hysterersis compensation.
-20
-10
0
10
20
Magnitude (dB)
10
-1
10
0

10
1
-135
-90
-45
0
45
90
Phase (deg)
Bode Diagram
Frequency (Hz)

H. 10 Nonlinear ‘lead-lag’ like compensation
The pressure loop is highly nonlinear with respect
to the position amplitude. The nonlinearity of the
pressure loop is due to the nonlinear process of the
pressure buildup as well as due to the effect of the
position on the pressure loop. Fig. 8a shows that
(i) there is a delay in the pressure loop, and (ii)
complex interactions in the pressure loop decrease
the damping of this loop. The system delay in
connection with the phase lag potentially pushes
the system into the unstable region. When the HC
is in function, the pressure loop performs quite
stable with an improvement of the damping and an
additional phase lead. The pressure loop response
with HC especially shows the effectiveness of the
compensation regardless of the different
amplitudes (Fig. 8b). This could benefit the
hysteresis compensation as it is a kind of position-

based compensation. Fig.9 is withdrawn from Fig.
8 with a view to look closer to a typical case where
the HC plays very well its role. This comparison
shows that the hysteresis compensation
dramatically improves control performance in
adding a phase lead to the system as well as
providing damping to the pressure loop. The
action of the hysteresis compensator is somehow
similar to lead-lag compensation (Fig. 10).
However, this is a kind of nonlinear ‘lead-lag’
compensator with varying parameter.
5. Conclusion
The MS model of the extracted hysteresis is
simple and well-suited to build in the control
scheme, in which the PID feedback linearizes the
system, whereas the hysteresis compensation
functions effectively as a feedback compensator
providing suitable damping and phase lead for the
pressure loop at any positions.
The nonlinear effects of hysteresis lead to a
complex interaction between the pressure loop and
the joint position. An FRF analysis does not
provide an explicit explanation for the hysteresis
effects in the system, but helps to see the
effectiveness of the compensation, so-called
nonlinear lead-lag compensator.
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Elsevier, 2010,20(3),402-414
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Vo Minh Tri, born in Bentre (1970),
obtained his Ph.D from Katholieke

Universiteit Leuven, Belgium in
2010. Formerly, graduated as
Agricultural Mechanical Engineer at
Cantho University, and received
M.Sc. degree in Mechanical Engineering at
National University of Ho Chi Minh, Vietnam in
1993 and 1998 respectively. Currently, he is senior
lecturer at Cantho University, Department of
Automation Technology where he is involved in
teaching Mechatronics Design, Measurement
systems, Systems Engineering. His main research
interest concerns the implementation of
mechatronic into agricultural production.