Proceedings of the 2020 International Conference on Advanced Mechatronic Systems, Hanoi, Vietnam, December 10 - 13, 2020
A Modified Bouc-Wen Model of Pneumatic
Artificial Muscles in Antagonistic Configuration
Quy-Thinh Dao∗ , Hai-Trieu Le, Manh-Linh Nguyen, Trong-Hieu Do, Minh-Duc Duong
Department of Industrial Automation
Hanoi University of Science and Technology
Hanoi, Vietnam
thinh.daoquy@hust.edu.vn
Abstract—Pneumatic Artificial Muscles (PAMs) exhibit hysteresis, nonlinear, and uncertain characteristics. This makes it
difficult to build a highly precise model of PAM. In this study, a
modified Bouc-Wen model is used to describe the hysteresis in the
relationship between internal pressure and joint angle of PAMs in
the antagonistic configuration. Through an identification method
using particle swarm optimization, it is verified that the proposed
model can approximate the behavior of PAM based antagonistic
actuator with high accuracy. This could lead to enhancing the
tracking performance of PAMs.
Index Terms—Pneumatic artificial muscles, hysteresis model,
Bouc-Wen model, system identification, antagonistic muscles
I. I NTRODUCTION
This study aims at presenting a modified Bouc-Wen (MBW)
model for describing the behavior of PAMs in the antagonistic
setup. In the proposed model, the additional input difference
term is added to adapt to the different behavior of PAMs during
contraction and relaxation processes. The proposed MBW
model is validated by identification method using particle
swarm optimization. To the end, this paper is organized
as follows. Section II provides the background of the BW
model and the proposed modified one. Then, the experimental
apparatus and data collection for parameter identification are
described in Section III. Next, the procedure for obtaining
the model parameters is presented in IV. Finally, Section V
concludes the paper.
Pneumatic Artificial Muscle (PAM), a soft actuator has
attracted great attention by many researchers in the last decade
[1]–[7]. In comparison to electrical motors and hydraulic
actuators, PAM performs two significant advantages which are
very high power/weight and power/volume ratios [8]. Besides,
PAM has many advantages such as lightweight, flexibility, etc.
Especially, PAM acts similarly to a human skeletal muscle.
When supplied with the air, PAM contracts in the longitudinal
direction and enlarged in the radial direction, and vice versa.
Thus, PAMs are widely applied to human-interact applications. A single PAM is often used in longitudinal direction
applications such as Body Weight Support systems. PAMs in
the antagonistic configuration are usually applied in rotation
applications as medical appliances, biorobots, arm orthotics,
lower-limb rehabilitation, and so forth. However, the performance of PAM-based systems is limited by the nonlinear input
pressure–length relationship and by the inherently complex
hysteresis. It leads to difficulty in both model and control
of PAM. It also makes the low performance of PAM-based
systems.
In order to describe the hysteresis behavior of PAM in both
single-mass and antagonistic systems, a number of hysteresis
models have been employed such as Prandtl–Ishlinskii [3],
Maxwell slip model [5], [6], Preisach model [7], and BoucWen model [10], [11]. One of the most widely adopted ones
is the Bouc–Wen (BW) hysteresis model whose best feature
is its versatility.
where y(t) denotes the output of the BW model, u(t),
and ω(t) denote the applied input and the hysteresis state,
respectively; ρ ≤ 1 is the weighting parameter; k is the
stiffness coefficient; and ε, ψ, γ and n ≥ 1 are the parameters
which govern the shape and amplitude of the hysteresis curve.
Alternatively, the BW model in equation (1) is rewritten as
follows:
y(t) = ρk + k(1 − ρ) ε − ψsgn(u̇)|Γ|n−1 − γ|Γ|n u̇(t)
(2)
y − ρku
where Γ =
k(1 − ρ)
In the case of n = 1, k = 1, ρ = 0, equation (2) is expressed
as:
y(t) = εu̇(t) − ψ|u̇(t)|y(t) − γ u̇(t)|y(t)|
(3)
This research is funded by the Hanoi University of Science and Technology
(HUST) under project number T2020-TT-002.
It is a special case of the BW model which is considered
in this paper. For further developments, the relationship (3)
II. M ATHEMATICAL F ORMULATION OF M ODIFIED
B OUC -W EN M ODEL
First, the formulation of the original BW model is provided
together with its discretizing procedure. After that, the proposed modification BW model is presented. The original BW
model is expressed as equation (1) [9]:
y(t) = ρku(t) + (1 − ρ)kω(t)
ω̇(t) = εu̇(t) − ψ|u̇(t)||ω̇(t)|n−1 ω(t) − γ u̇(t)|ω̇(t)|n
ω(0) = ω0
(1a)
(1b)
(1c)
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can be discretized base on Taylor Series expansion method as
follows.
The first order derivative can be simply approximated as
∆t2
(4)
2
where σ ∈ (t, t + ∆t), ∆t is the sampling time, t = k∆t is
the sampling instant. If y(t) 6= 0 , the first order derivative
ẏ(t) is obtained
y(t + ∆t) = y(t) − ẏ(t)∆t + ÿ(σ)
y(t + ∆t) − (1 − µ1 )y(t)
(5)
∆t
ÿ(σ) ∆t2
is defined as a higher order positive
in which µ1 =
y(t) 2
function of ∆t. If the sampling period ∆t is sufficiently small,
(1 − µ1 ) will be closely equal to 1. Thus, the derivative ẏ(t)
can be derived from (5) as:
ẏ(t) =
y(t + ∆t) − λ1 y(t)
(6)
∆t
where λ1 is a parameter which is closely equal to 1 when
∆t is small enough. We can easily see that equation (6) is also
valid even if y(t) = 0.
Similarly, we have the approximation of derivative u̇(t) as
ẏ(t) ∼
=
u(t + ∆t) − λ2 u(t)
(7)
u̇(t) ∼
=
∆t
where λ2 is a parameter which is closely equal to 1 when ∆t
is sufficiently small.
From (3), (6), and (7), we have
yk = λ1 yk−1 + ξϑk − ψ|ϑk |yk−1 − βϑk |yk−1 |
(8)
denote k∆t as k and define
uk − λ2 uk−1
ϑk =
, ξ = ε∆t, ψ = Ψ ∆t, β = γ∆t (9)
∆t
Relation (8) is the numerical approximation of the BoucWen model in the discrete-time domain. The Bouc-Wen model
40
30
is rate-independent, i.e, limited to decribing invariant hysteresis curves regardless the increment/decrement of the input
frequency. This behaviour can be clearly in Fig. 1. Besides,
hysteretic effects found in most smart materials, especially in
pneumatic artificial muscle is rate-dependent. According to the
our research, there are a number of extensions of BW model
which have been reported in the literature typical as [13], [14].
Nonlinear terms are included in the standard BW form not only
depend on the sign of the input derivative, but also on the input
itself. This extension offers greater flexibility in shape control
and has the ability to depict asymmetric hysteresis curves with
time invariant parameters. Meanwhile, the proposed extension
[14] is essentially based on the addition of a function with no
further nonlinear memory, i.e. , a polynomial input function
created cascading with the BW model the standard has the
local Lipschitz continuity attribute. It can be seen that both of
these extensions require meticulous methods of determination
to yield satisfactory results In order to describe the hysteresis
behavior of PAM according to the change of its pressure,
equation (8) is further modified by introducing the first-order
input difference term νu,k which is define as
νu,k = |uk − uk−1 |
(10)
Then, the proposed modified-BW (MBW) model is given
as follows
yk = λ1 yk−1 + (ξ1 + ξ2 νu,k ) ϑk
− (ϕ1 + ϕ2 νu,k ) |ϑk |yk−1
− (β1 + β2 νu,k ) ϑk |yk−1 |
(11)
III. E XPERIMENTAL A PPARATUS AND DATA C OLLECTION
A. Experimental Apparatus
The overall experiment apparatus is shown in Fig. 2, which
consists of two PAMs, two electric control valves, a potentiometer, and a control system, etc. (Table I). In this research,
two PAMs with 25 mm of diameter and 500 mm of length
are connected through a pulley to become the antagonistic
configuration (Fig. 3a). As shown in Fig. 3b, the PAM is
fabricated at our laboratory which has an inner rubber tube and
a layer of non-expendable braided shell. The tube is wrapped
in a double-helix-braided shell woven at a predetermined
20
10
0
-10
-20
-30
-40
-1
-0.5
0
0.5
1
Fig. 1. Comparison of input-output relation described by the Bouc-Wen model
(λ1 = 0.995, λ2 = 0.995, ξ = 0.8, ψ = 0.04, and β = −0.08) in the
discrete-time domain at different frequencies.
Fig. 2. Experiment Apparatus.
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Triangle signal: The control signal has the triangle shape
with 0.2 MPa amplitude and frequency varies from 0.2
to 0.8 Hz.
The experimental data, including the control signal, the measured angle of the actuator, and the pressure are collected for
further analysis. The sampling time is set as Ts = 5 ms. The
first data of control signal is used to identify the MBW model
parameters. Two others are used for model validation.
TABLE I
C OMPONENTS OF EXPERIMENT SYSTEM
Components
PAM
Potentiometer
ECV
Controller
Model
WDD35D8
ITV-2030-212S-X26
Myrio-1900
•
Branch
HUST
Zhejiang
SMC
National Instruments
IV. M ODEL I DENTIFICATION P ROCEDURE
A. Particle Swarm Optimization
(a)
(b)
Fig. 3. (a) The antagonistic setup and (b) the fabricated PAMs in experiments.
angle. Both ends of the rubber tube are closed with caps.
The joint angle is measured by a potentiometer WDD35D8
from Zhejiang, China. The pressure inside of each PAMs are
manipulated by electric control valves ITV-1030-04N2OL5 by
SMC company, Japan. The control system includes a personal
computer and a realtime controller Myrio-1900 from National
Instrument (USA). The Myrio is connected to the potentiometer via analog input module and regulates the pressure
supplying to PAMs via analog output channel.
The MBW model in section II includes many parameters for
constructing the shape of the hysteresis curve. The parameters
of the proposed model can be estimated through an optimization tool. In this research, the particle swarm optimization
method is employed to identify the model parameters. The
procedure of the original PSO can be described as following
steps [12]:
• Initialize the time to zero and set a number for initial
i,d
position xi,d
0 and initial velocity ν0
i,d
• Evaluate the fitness of each particle F xk
.
•
•
B. Data Collection
In order to identify the model parameters, the input
(control signal) and output (joint angle) must be collected
first. After that, an optimization method is used to identify
the model parameters. The input-output data is collected by
two following steps:
Step 1: The initial joint angle is set at 0 degrees by
supplying the same nominal pressure (P0 ) to both PAMs.
This value will determine the joint compliance and in this
paper P0 is set as 0.2 MPa.
Step 2: The joint angle can be changed by sending a
control signal to the electrical control valves. This signal is
proportional to the different pressure inside each PAM. Three
types of control signals are used for identification:
• Sine wave: The sinusoidal signal with 0.2 MPa amplitude
and frequency varies from 0.2 to 0.8 Hz.
• A mixed-frequency sine wave with both various amplitude and frequency, as in the following equation:
u(k) = Asin(2πf t) + 0.6Asin(2π0.3f t)
+0.7Asin(2π1.5f t) + 0.1Asin(2π4f t)
(12)
in which A = 0.1 MPa is the basic amplitude, and f =
0.5 Hz is the basis frequency of the input signal.
•
•
Set the P bi,d
k to the

P bi,d
k−1
P bk =
xi,d
F
k
better performance as follows
F xi,d
≥ F P bi,d
k
k−1
i,d
xi,d
<
F
P
b
k
k−1
Set the Gbi,d
k to the position of particle with the best
fitness within the swarm as
1,d
1,d
n,d
Gbi,d
k ∈ Pbk−1 , P bn
k−1 , · · · P bk−1
o
i,d
1,d
n,d
F Gbk = min F (P bk−1 ), F (P b1,d
k−1 ), · · · F (P bk−1 )
Update the velocity vector for each particle according to
the following
rule

V
,
if νki,d ≥ Vmax

max


i,d

−Vmax ,
if νk < −Vmax
i,d
νk+1 =
i,d
Iω νki,d + p1 r1 P bi,d

k − ik


otherwise

+ p r Gbi,d − ii,d ,
2 2
k
k
Update the position of each particle according to
i,d
i,d
xi,d
k+1 = xk + vk+1
• Let k = k + 1
i,d
• Compute the new F (xk ) until the iteration to be terminated or the least value for F to be achieved.
where Iω is inertia weight, p1 is cognitive learning gain, p2
is social learning gain; r1 and r2 are random numbers in the
range of [0,1]; P bi,d
k is the best known position along the d th
dimension of particle i in iteration k; Gbi,d
k is the global best
known position among all particles along the d th dimension in
iteration k; and k = 1, 2, · · · , N denotes the iteration number;
N is the maximum allowable iteration number, Ns is the
population size. In addition, Vmax is the maximum velocity
evolution which is usually selected to be half of the length of
the search space.
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TABLE II
PARTICLE S WARM O PTIMIZATION PARAMETERS .
40
Iω max
0.9
Iω min
0.2
p1
1.5
p2
2.5
N
1000
Ns
60
4
20
2
0
0
-20
TABLE III
PARAMETERS OF P ROPOSED B OUC -W EN M ODEL .
-2
-40
0
λ1
0.9876
ϕ1
-0.0045
λ2
0.8542
ϕ2
-0.5068
ξ1
0.0016
β1
0.0080
ξ2
0.3372
β2
-0.2762
5
10
0
5
10
40
20
0
-20
B. Model Validation
By using PSO method, eight parameters of the proposed
MBW model are identified with the applied pressure is a sine
wave with 0.5 Hz frequency first. Then, the proposed model
is verified with two other input signals including triangle and
mixed-frequency sinusoidal waves. The chosen parameters of
PSO identification are provided in Table II. Figure 4 indicates
hysteresis curves collected in the experiment and estimation
curve obtained from the MBW model (11) with the parameters
in Table III at 0.5 Hz of sinusoidal signal frequency. Figure 5
and 6 show the model validation with the inputs are triangle
and mixed frequency sinusoidal signals, respectively. The root
mean square values of estimation errors between measured
data and estimated data of MBW are provided in Table IV.
It is verified that the MBW depicts a good approximation of
hysteresis characteristic of PAMs in an antagonistic setup with
the RMSE less than 4.00◦ .
-40
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Fig. 5. Model validation with input is the 0.5Hz triangle signal: (a), (b) are
measured and estimated values, and their deviation. (c) is the comparison of
input-output relationship between collected data (blue line) and MBW model
(dash-red line).
10
50
5
0
0
-50
-5
-100
-10
0
5
10
0
5
10
50
40
4
0
20
2
-50
0
0
-20
-100
-2
-40
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
0
5
10
0
5
10
Fig. 6. Model validation with input is the mixed-frequency sinusoidal signal:
(a), (b) are measured and estimated values, and their deviation. (c) is the
comparison of input-output relationship between collected data (blue line)
and MBW model (dash-red line).
40
20
0
-20
-40
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Fig. 4. Estimation results of the 0.5Hz sinusoidal input signal: (a), (b) is
measured and estimated values, and their deviation. (c) is the comparison of
input-output relationship between collected data (blue line) and MBW model
(dash-red line).
M ODEL ACCURACY
IN
TABLE IV
T ERM OF RMSE VALUE B ETWEEN M EASURED
AND E STIMATED DATA .
Input Signal
Sine wave
Triangle wave
Mixed frequency signal
Estimation Accuracy
0.2 Hz
0.5Hz
0.8541
1.3187
0.9422
1.1516
3.5095
(RMSE)
0.8Hz
1.3253
1.6288
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V. C ONCLUSIONS
This research proposed a modified Bouc-Wen model to describe the behavior of PAMs in an antagonistic configuration.
By implementing the input different term, the MBW model is
able to characterize the rate-dependent input-output hysteresis
relation of PAMs with good accuracy. Particularly, the MBW
model is designed in the discrete-time domain, it is facilitate to
implemented in any real applications. The MBW exhibits not
only promising modeling and characterization but also toward
high tracking performance in PAM-based applications.
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