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IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020
Fiber-Reinforced Soft Bending Actuator Control
Utilizing On/Off Valves
Cong Chen, Wei Tang, Yu Hu, Yangqiao Lin, and Jun Zou
Abstract—Soft fluid-driven robots have great potential for safe
interaction with humans, and for adaptation to complex and unpredictable environments with the compliance brought about by soft
materials. However, the respective complex structure and massive
use of nonlinear and viscoelastic soft materials, and the nonlinear
fluid-driven dynamics, result in the nonlinear dynamic behavior
of soft robots, and thus pose great challenges to system modeling
and dynamic control. In this study, a complete model is first established by considering the nonlinear behavior of fiber-reinforced soft
bending actuator (FRSBA) and the nonlinear dynamics of pneumatic system regulated by two high-speed on/off valves. Notably,
the nonlinear dynamic behavior of FRSBA is first identified as
parametric uncertainties using the multi-sine pressure excitation
signal, and the effects of nonlinear pneumatic system are fully taken
into account. Then, an adaptive robust controller is designed to
handle the system nonlinearities with a guaranteed transient and
steady-state performance. Finally, the comparative experiments
demonstrate the effectiveness of proposed modeling method and
high performance of adaptive robust controller
Index Terms—Modeling, control, and learning for soft robots,
motion control, dynamics, system identification, adaptive robust
control.
I. INTRODUCTION
HE benefit of the inherent compliance brought about by
soft materials is that soft robots have, compared with conventional rigid robots, the innate advantages of safe interaction
with humans and adaptability to complex and unpredictable
environments at potential low cost [1], [2]. Over the past decade,
many studies have carried out meaningful work in this field, including design, manufacturing, sensing, modeling, and control.
Consequently, several soft robots with various functions, such
as twisting, bending, stretching, rotation, have been developed
based on various approaches, including shape memory alloys,
shape morphing polymers, dielectric elastomers, tendon drive,
piezoelectric actuation, and fluid power [3], [4].
Amongst these developments, fluid-driven soft robots, most
of which are pneumatically driven, have the advantages of
T
Manuscript received July 3, 2020; accepted July 27, 2020. Date of publication August 7, 2020; date of current version August 28, 2020. This letter
was recommended for publication by Associate Editor C. Laschi and Editor
M. Cianchetti upon evaluation of the Reviewers’ comments. This work was
supported by the National Natural Science Foundation of China under Grants
51875507 and 51821093. (Corresponding author: Jun Zou.)
The authors are with the State Key Laboratory of Fluid Power &
Mechtronic Systems, School of Mechanical Engineering, Zhejiang University,
Zhejiang 310027, China (e-mail: congchen@zju.edu.cn; weitang@zju.edu.cn;
11725056@zju.edu.cn; 11325035@zju.edu.cn; junzou@zju.edu.cn).
This letter has supplementary downloadable material available at http://
ieeexplore.ieee.org, provided by the authors.
Digital Object Identifier 10.1109/LRA.2020.3015189
large deformation/force, good power-to-weight ratio, and low
manufacturing cost. Additionally, they have received extensive
attention, and are typically represented by a fiber-reinforced soft
bending actuator (FRSBA) [5]. However, the massive use of
multiple nonlinear, viscoelastic soft materials leads to hysteresis
and the nonlinear dynamic behavior of soft robots. These issues,
along with the lack of precise and integrated sensors, nonlinear fluid dynamics, and the corresponding complex structure,
increase the difficulty of dynamic control for fluid-driven soft
robots [6], [7]. Hence, a unified framework for the control of
soft robots is still lacking [8], most fluid-driven soft robots use
open-loop control [9].
Despite the enormous challenges, various attempts have been
made to develop a close-loop control for fluid-driven soft robots.
A few studies have attempted to develop a low-level controller
for fluid-driven soft robots by dynamically controlling the inner
pressure [10] or fluid mass [11] in the robot chamber, instead of
the position. By ignoring the robot dynamics, the control duty
is simplified, but the control performance cannot be guaranteed. Various studies have used a model-free tuning or training
approach. Gerboni et al. [12] designed a proportional-integral
(PI) controller with a low-pass filter for flexible fluid actuators
(FFAs). Wu et al. [13] used radial basis function (RBF) neural networks to train the turning ability of a differential drive
pneumatic soft robot. These studies have introduced a relatively
simpler method for fluid-driven soft robot control, but the stability analysis and convergence proofs are difficult. Moreover,
some studies have attempted to involve the theoretical kinematic
and dynamic models in the controller design. Polygerinos et al.
[14] established a quasi-static analytical FRSBA model based
on the incompressible Neo-Hookean (HK) material model, and
used it as an angle filter in feedback control. Bieze et al. [15]
proposed a real-time numerical integration strategy based on
the finite element method (FEM) to obtain the quasi-static
forward kinematic model (FKM) and inverse kinematic model
(IKM) of the compact bionic handling assistant, and design a PI
controller based on the state estimator of the FKM simulation.
Marchese et al. [16] derived a dynamic model from the constant
curvature model and obtained the optimal reference inputs to
the actuator to realize the initial trajectory. Falkenhahn et al.
[17] established a dynamic bionic handling assistant model
based on the concentrated mass model and Euler-Lagrange
method, and used feedback linearization to design a cascade
controller, whose inner and outer loop are a proportional (P) and
proportional-differential (PD) controller combined with a feedforward controller, respectively. These studies investigated the
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CHEN et al.: FIBER-REINFORCED SOFT BENDING ACTUATOR CONTROL UTILIZING ON/OFF VALVES
feasibility of fluid-driven soft robot nonlinear behavior control.
However, the abovementioned kinematic and dynamic models
are imprecise, complex, and difficult to analyze, which increases
the complexity of controller design. Moreover, other studies
have attempted to control fluid-driven soft robots based on an
empirical model. Li et al. [18] proposed a Jacobian estimation
algorithm based on an adaptive Kalman filter and designed a
controller by solving the linear state equation for continuum
robots. Similarly, Zhang et al. [19] predicted the Jacobian based
on real-time FEM computation, and designed a pseudo-inverse
controller. By estimating the image Jacobian and generating
the reference pressures for the inner-loop proportional-integraldifferential (PID) pressure controllers, Greer et al. [20] introduced the eye-in-hand visual servo control for soft continuum
robots. Similarly, Fang et al. [21] learned the inverse mapping of
the visual servo and designed a controller based on the localized
Gaussian process regulation (LGPR) for a soft manipulator.
Hyatt et al. [22] formulated a linearized discrete state space
representation based on gradients used within a neural network
and developed a model predictive controller (MPC). Skorina
et al. introduced a second-order system for antagonistic soft
pneumatic actuators by fitting the theoretical response with an
experimental step response, and designed an iterative sliding
mode controller [23] and a model reference adaptive controller
[24]. Similarly, Wang et al. [25] estimated the parameters of a
second-order model at different frequencies, and first established
the stability analysis and convergence proof with a robust backstepping controller. These studies provide a simpler approach
toward system modeling and controller design compared with
the strategies involving more complex theoretical models. However the nonlinear behaviors are not perfectly described and the
validities are limited to the specified experimental conditions
[26].
In this study, the empirical model approach was used to
describe the nonlinear dynamic behaviors, and system identification based on multi-sine pressure excitation was introduced
to identify the dynamic model of a fluid-driven soft actuator.
Additionally, the nonlinear behaviors were synthesized into
parametric uncertainties. To measure the bending angle of the
actuator, a commercial inclinometer was mounted onto the end
position. Moreover, the nonlinear model of the pneumatic system regulated by two high-speed on/off valves was analyzed by
considering the dead-zone and nonlinear relationship between
duty cycle input and mass flow rate of the valve actuated by the
PWM signals. Subsequently, an adaptive robust controller was
designed to handle the system nonlinearities and ensure satisfactory transient performance. Finally, the proposed modeling
method and controller were validated through bending angle
trajectory tracking experiments.
II. PROBLEM FORMULATION AND SYSTEM MODELING
The investigated pneumatic FRSBA control system is shown
in Fig. 1. The compressed air source was regulated using a
precision pressure regulator (FESTO LRPS). Two high-speed
on/off solenoid valves (MAC BV108) were employed to control
the flow rates of the air charging and exhaust, respectively.
Fig. 1.
Schematic of FRSBA control system.
Fig. 2.
Structure of FRSBA with inclinometer.
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The FRSBA inner chamber pressure and regulated air source
pressure were measured by the pressure sensors (FESTO SPAN).
An inclinometer (Wit-Motion WT931) was mounted onto the
end position to detect the bending angle, as shown in Fig. 2. The
control algorithm was developed using the LabVIEW software
and executed in the NI CompactRIO-9047 controller.
A. FRSBA Dynamics
Because theoretical models are always complex and difficult
to analyze, the complexity of the controller design increases. An
empirically established model was introduced to describe the
FRSBA dynamics, for which a key step is to select an effective
excitation signal. Because the inner pressure Pv of the FRSBA
is a positive continuous physical state, which cannot transform
arbitrarily fast, and because the excitation signal must comprise
sufficient frequency components in an appropriate frequency
band, the multi-sine excitation signal Pvd satisfying the abovementioned objectives can be easily designed as follows:
Pvd =
N
Ak sin (2πfk t + φk ) + B k = 1, 2, . . . , N
k=1
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IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020
TABLE I
PARAMETERS OF IDENTIFIED MODEL
can be seen in Appendix A), as follows:
θ(s)
b0
= 2
Pv (s)
s + a1 s + a0
Fig. 3. (a) [0, 1] Hz excitation signal tracking result; (b)[0, 0.5] Hz excitation
signal tracking result; (c) Identified model output validation under [0, 1] Hz
excitation signal; (d) Identified model output validation under [0, 0.5] Hz
excitation signal.
where fk , k = 1, 2, . . . , N are the consecutive frequencies distributed in the frequency range of interest; N is the number of
harmonics; Ak , φk are the amplitude and phase shift angle of
the kth harmonic, respectively; B is an additional offset to ensure
a positive excitation signal value.
Considering the shortage of prior knowledge with regard
to the FRSBA dynamics, it should be assumed that all power
spectrum components are equally important [27] such that Ak
can be expressed as follows:
A
Ak = √
N
(2)
where A is the amplitude of Pvd .
To increase the efficiency of Pvd , the phase shift angle φk can
be designed by minimizing the relative crest factor, as follows:
RP F (Pvd ) =
[max (Pvd ) − min (Pvd )]
√
2 2rms (Pvd )
(3)
To identify the dynamic FRSBA model, the inner pressure
tracking controller proposed in step 2 of the controller design
section was firstly used to make the inner pressure Pv track
the desired excitation signal Pvd , as closely as possible. As
shown in Fig. 3(a), good inner pressure tracking performances
were achieved when the frequency range of Pvd is [0, 1] Hz.
In this experiment, inner pressure Pv and the FRSBA bending
angle were measured and recorded as the excitation input and
measured output, respectively, thus the identification data (Data
1) of 120 seconds were obtained. In the same way, another set of
identification data (Data 2) of 60 seconds were obtained when
the frequency range of Pvd is [0, 0.5] Hz and the tracking result
can be seen in Fig. 3(b).
Synthesizing the complexity of model and output fitting accuracy, a second-order transfer function model was established
based on the first 60 seconds of Data 1 in MATLAB (more details
(4)
Before the identified model can be used for bending angle
trajectory tracking control, it was validated with the second
60 seconds of Data 1. As shown in Fig. 3(c), the identified
model with the mean parameters shown in the Table I is feasible
with a NRMSE fitness value of 88.44%. To further illustrate
the effectiveness of the identified model, Data 2 were also
used for validation in Fig. 3(d), and the resulting NRMSE
fitness value is 89.82%. To fully consider the dynamic behavior
of FRSBA, the 99.7% confidence intervals of the parameters
shown in Table I were estimated, and considered as bounded
parametric uncertainties. The corresponding confidence interval
of the model output is indicated by the light magenta region in
Fig. 3(c) and (d). The validation results demonstrate that the
identified model with parametric uncertainties can fully capture
the FRSBA dynamics.
B. Pneumatic System Dynamics
The air dynamics of the FRSBA inner pressure Pv during
FRSBA bending can be treated as an isothermal process. Because the FRSBA is reinforced by soft fiber, the inner chamber
deformation is restricted as a result, and the inner chamber
volume V does not change substantially during FRSBA bending.
Hence, it is reasonable to consider V as a volume parameter with
parametric uncertainty. Then, the dynamics of Pv can be easily
derived from the ideal gas law, as follows:
Ṗv =
RT
(Qm1 − Qm2 )
Sp V
(5)
where R = 287N m/(Kg · K) is the ideal gas constant; T is the
absolute air temperature; Sp = 1000 is a constant coefficient;
Qm1 and Qm2 are the mass flow rate of the air charging and
exhaust, respectively.
The high-speed on/off valves are actuated by pulse width
modulation (PWM) signals with a carrier frequency of 60 Hz.
Depending on the level of the pulse signal, the valve will work
in the full open or closed state, and the full open flow rate Qme
can be expressed as follows [28]:
⎧
P
, PPud ≤ 0.528
⎨Cf Av C1√Tu
1
κ−1
Qme (Pu ,Pd) =
⎩Cf Av C2√Pu Pd κ 1− Pd κ , Pd > 0.528
T Pu
Pu
Pu
(6)
where Pu and Pd are the upstream and downstream pressures, Cf is a non-dimensional, discharge coefficient, Av is
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CHEN et al.: FIBER-REINFORCED SOFT BENDING ACTUATOR CONTROL UTILIZING ON/OFF VALVES
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C. Problem Formulation
From Equations (4), (5), (7), (8), the entire system dynamics
can be synthesized to simplify the controller design by separating the air charging and exhausting process, as follows:
Fig. 4. Experimental data of duty cycle ui and compensated virtual duty cycle
input vi under different input and output pressure conditions (colored solid
lines with point markers); and compensation model with mean parameters (blue
dash-dot line).
the valves orifice area, κ=1.4 is the heat capacity ratio of air,
κ−1
κ
2
2κ
( κ+1
) κ+1 and C2 = R(κ+1)
are constants.
C1 = R
The calculable average mass flow rate value of Qmi is approximately regulated by changing the duty cycle of the PWM
signals, and can be generally expressed as a product of the duty
cycle ui and the full open flow rate Qme [29]. In this study, a
virtual duty cycle input vi was defined as the actual relationship
between the mass flow rate Qmi and the full open flow rate
Qme under different input and output pressure conditions for
the valves. Consequently, the average mass flow rate Qm1 and
Qm2 can be calculated as follows:
Qm1 = v1 Qme (Ps , Sp Pv + Pa )
Qm2 = v2 Qme (Sp Pv + Pa , Pa )
(7)
where Pa = 101325 P a is the standard atmosphere; Ps = Pa +
250000 P a is the source pressure.
According to the results obtained by quasi-static experiments
and presented in Fig. 4, the relationship between the virtual
duty cycle input vi and actual duty cycle input ui exhibited a
dead-zone and nonlinearity owing to the disregarded dynamic
characteristics of the valves, such as the response times. Hence,
model compensation for the duty cycle input ui was introduced
to handle these performance degradation issues, as follows:
vi =
kv (ui − udz ) if ui > udz
0
else
ẋ1 = x2
ẋ2 = −γ1 x1 − γ2 x2 + γ3 Pv
f (Pa , Ps , Sp Pv +Pa)(γ4 ui −γ5),if ui ≥ udz
Ṗv =
i = 1, 2
0
,else
(9)
where the states are x = [x1 , x2 ]T = [θ, θ̇]T ; the
model
parameters
are
γ =[γ1 , γ2 , γ3 , γ4 , γ5 ]T =
RT
T
K
,
K
u
]
;
i = 1, 2 represents the
[a0 , a1 , b0 , SRT
v Sp V
v dz
pV
air charging and exhausting, respectively; f (Pa , Ps , Pv + Pa )
is a known item expressed as follows:
Qme (Ps , Sp Pv + Pa ) , i = 1
−Qme (Sp Pv + Pa , Pa ) , i = 2
(10)
In actual operation, according to the abovementioned analysis, the model parameters γ cannot be precisely determined.
However, the parametric uncertainties are bounded, and the
parametric bound can be determined as follows:
f (Pa , Ps , Sp Pv + Pa ) =
Δ
γ ∈ Ωγ = {γ : γmin ≤ γ ≤ γmax }
(11)
where the lower bound γmin = [γ1 min , γ2 min , γ3 min , γ4 min ,
γ5 min ]T , and the upper bound γmax = [γ1 max , γ2 max ,
γ3 max , γ4 max , γ5 max ]T , are known.
III. CONTROLLER DESIGN
In this section, the adaptive robust controller (ARC) proposed
in [30] was designed to handle the systems nonlinearities and
ensure transient performance.
A. Discontinuous Projection Mapping
Discontinuous projection mapping was introduced to
guarantee the bounded parameter estimations. Let γ̂ =
[γ̂1 , γ̂2 , γ̂3 , γ̂4 , γ̂5 ]T denote the estimation of γ; the estimation
error is defined as γ̃ = γ̂ − γ = [γ̃1 , γ̃2 , γ̃3 , γ̃4 , γ̃5 ]T . A discontinuous projection can be defined as follows:
⎧
⎨ 0, if γ̂i = γi max and •i > 0
P roj γ̂ (•i ) = 0, if γ̂i = γi min and •i < 0
(12)
⎩
•i , otherwise
Accordingly, the adaptation law is expressed as follows:
i = 1, 2
(8)
where udz is the dead-zone of the duty cycle input, and kv is a
proportional gain. Based on the least square method, mean values of kv and udz are estimated from quasi-static experimental
data. Additionally, the bounded parametric uncertainties of kv
and udz are also estimated and introduced to cover all of the
valves input and output pressure conditions. The corresponding
confidence interval of the compensation model output is indicated by the light blue region in Fig. 4.
γ̂˙ = P roj γ̂ (Γτ )
(13)
where Γ > 0 is a positive-definite diagonal adaptation rate matrix, and τ is an adaptation function to be synthesized later. Thus,
it can be proven that, for any adaptation function τ [30], the
projection mapping expressed in Equation (13) has the following
properties:
P1 :
Δ
γ̂ ∈ Ωγ = {γ : γmin ≤ γ ≤ γmax }
P 2 : γ̃ Γ−1 P roj γ̂ (Γτ ) − τ ≤ 0, ∀τ
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IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020
Let us define a positive-definite function V1 = 12 S12 and the
input discrepancy S2 = Pv − Pvd , the following inequality can
be derived as (more details can be seen in Appendix B):
γ3
V̇1 ≤
K1 S12 + ε1 + γ3 S1 S2
(22)
γ3 min
TABLE II
WORKING MODE SELECTION STRATEGY
B. Working Mode Selection
To avoid frequent switching between the charging and exhaust
valve, the working mode selection strategy proposed in [31] was
implemented to select the working mode based on the desired
bending velocity θ̇d and trajectory tracking error of the bending
angle x1 − θd . As presented in Table III, Mode 1 and Mode 2
represent the air charging and exhaust, respectively, while Mode
3 is set to turn off both valves.
C. Adaptive Robust Controller Design
The two-step backstepping-based ARC controller is designed
as described below.
1) Step 1: Let e denote the bending angle tracking error of
FRSBA, that is e = x1 − θd ; a switching-function like quantity
is defined as follows:
S1 = ė + KS1 e
(15)
where KS1 > 0 is a positive gain to be selected. If S1 is small or
converges to zero, then the bending angle tracking error e will
also be small or converge to zero.
The dynamics of S1 can be derived from Equations (9) and
(15), as follows
Ṡ1 = ë + KS1 ė = −γ1 x1 − γ2 x2 + γ3 Pv + w
(16)
where w = KS1 (x2 − θ̇d ) − θ̈d is known, and the designed
control law of Step 1 can be expressed as follows:
Pvd = Pvda + Pvds1 + Pvds2
(17)
1
−ϕTa γ̂a − w
γ̂3
(18)
Pvda =
Pvds1 = −
1
γ3 min
K1 S 1
(19)
(i). S1 Pvds2 ≤ 0
(20)
where ε1 > 0 is a design parameter that can be arbitrarily
small; ϕb = [−x1 , −x2 , Pvda ]T , γ̃b =[γ̃1 , γ̃2 , γ̃3 ]T ; the adaptation function can be designed as follows:
τ1 = ϕb S1
Ṗvdc is the part that can be calculated of Ṗvd , Ṗvdu is the
incalculable part, and the designed control law of step 2 is
expressed as follows:
uida =
uid = uida + uids1 + uids2
(26)
γ̂5 f (Pa , Ps , Pa + Sp Pv ) + Ṗvdc − γ̂3 S1
γ̂4 f (Pa , Ps , Pa + Sp Pv )
(27)
uids1 = −
(21)
1
γ4 min
K2 S 2
(28)
where K2 > 0 is a positive gain to be selected, and uids2 is an
additional feedback item that must satisfy the following robust
performance conditions:
(i). S2 f (Pa , Ps , Pa +Sp Pv )uids2 ≤ 0
(ii). S2 γ4 f (Pa , Ps , Pa +Sp Pv )uids2 −ϕTc γ̃ ≤ ε2
(29)
∂Pvd
T
vd
where ϕc = [−∂P
∂x2 ϕa , S1 − ∂x2 Pv, f (Pa, Ps, Pa +Sp Pv )uida,
T
−f (Pa , Ps , Pa + Sp Pv )] , and the adaptation function can be
designed as follows:
τ2 = τ1T , 0
where K1 > 0 is a positive gain to be selected, ϕa =
[−x1 , −x2 ]T , γ̂a = [γ̂1 , γ̂2 ]T , and Pvds2 is an additional feedback item that needs to satisfy the following robust performance
conditions:
(ii). S1 γ3 Pvds2 − ϕTb γ̃b ≤ ε1
2) Step 2: In step 2, the control input ui is synthesized to
make the input discrepancy S2 of step 1 converge to zero or
bring it within some tolerance range with a guaranteed transient
performance. For the air charging (Mode 1, i = 1) or exhaust
(Mode 2, i = 2) process, the input discrepancy dynamics can be
synthesized from Equations (9), (17), (18), (19), (20), as follows:
Ṡ2 = f (Pa , Ps , Pa +Sp Pv ) (γ4 ui −γ5 )− Ṗvdc + Ṗvdu
(23)
where
∂Pvd ˙
∂Pvd
∂Pvd ˆ
∂Pvd
Ṗvdc =
+
x2 +
(24)
ẋ2 +
γ̂
∂x1
∂x2
∂t
∂γ̂a a
∂Pvd ˆ2
ẋ2 − ẋ
(25)
Ṗvdu =
∂x2
T
+ ϕc S 2
(30)
Let us define a positive-definite function V2 = V1 + 12 S22 , the
following inequality can be derived as (more details can be seen
in Appendix B):
V˙2 ≤ −λV2 + ε
(31)
where λ = min{K1 , K2 }, ε = ε1 + ε2 .
Therefore, the overall closed-loop system is stable and all
physical signals are bounded. Additionally, the trajectory tracking is guaranteed to have a prescribed transient and steady-state
performance, which can be quantified as follows:
ε
[1 − exp (−2λt)]
(32)
V2 ≤ V2 (0) exp (−2λt) +
2λ
Now consider the theorem of discontinuous projection based
ARC arguments. Let us define a positive-definite function V3 =
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CHEN et al.: FIBER-REINFORCED SOFT BENDING ACTUATOR CONTROL UTILIZING ON/OFF VALVES
TABLE III
CONTROLLER PARAMETERS
˙ under the assumption that
V2 + 12 γ̃ T Γ−1 γ̃, noting that γ̃˙ = γ̂,
the system is subjected to parametric uncertainties only, we can
derive that (more details can be seen in Appendix B)
γ4
γ3
−K1 S12 +
−K2 S22
(33)
V̇3 ≤
γ3 min
γ4 min
Therefore, the proposed ARC controller can guarantee that
S1 asymptotically converges to zero, thus the tracking error e
will also converge to zero, i.e., e → 0 as t → ∞.
IV. EXPERIMENTAL RESLUT
A. Experimental Setup
The effectiveness of the designed controller was verified
through experiments with the FRSBA control system, as shown
in Fig. 1. To show the advantages of proposed control method,
three comparative control methods were designed, as follows:
C1: PID Controller is used in C1;
C2: Sliding Mode Control (SMC) (or variable structure control, VSC) [32], the controller design of C2 is same to the
ARC in C3 but without using parameter adaptation, i.e., Γ =
diag{0, 0, 0, 0, 0};
C3: Adaptive Robust Control (ARC), the controller was designed in Section III;
and four reference trajectories were selected, as follows:
T1: 2 Steps type point-to-point trajectory;
T2: 3 Steps type point-to-point trajectory;
T3: Sinusoidal trajectory of 0.5 Hz;
T4: Multi-sine trajectory of [0, 0.5] Hz.
The sampling time was set to 0.005 s, and the controller
parameters used in the trajectory tracking experiments can be
seen in Table III.
The PID (C1) controller was first auto-tuned using the relay
feedback auto-tuning method, and then refined manually. For
reference trajectories T1 and T2, the proportional gain KP and
integral gain KI were larger to enhance the implementation of
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response and shorten the quasi-steady state error, in contract,
the effect of differential item is more obvious when reference
trajectory is T3 or T4, so that the phase lag of the response can
be reduced.
The SMC (C2) controller designed in this letter is a kind
of smoothed SMC law, in which, ε1 and ε2 are introduced to
overcome the chattering problem of ideal SMC law, and its only
drawback is a relatively larger steady state tracking error [33].
To simplify the controller parameter tuning, it is set that K1 =
K2 = KS1 = K. In general, the tracking error can be made as
small as possible by increasing K and/or decreasing ε1 and ε2 .
On the basis of smoothed SMC law, considering that steady
state tracking error is proportional to the model uncertainties [33] (that is γ̃, in this letter), ARC (C3) introduced
the parameter adaptation law to reduce γ̃. To ensure the effect of parameter adaptation, the adaptation rate matrix Γ =
diag{Γ1 , Γ2 , Γ3 , Γ4 , Γ5 } should not be too small, but a too large
Γ will lead to excess oscillating transient response due to the
measurement noises (especially the velocity and pressure).
B. Performance Index
To quantitatively compare the tracking results, the following
performance indexes introduced in [34] will be analyzed in each
experiment.
2
•L2 [e] = N1 N
k=1 |e(k)| , a scalar valued L2 norm, is used
as an objective numerical measurement of average tracking
performance for entire error sequence e(k), k = 1, 2, . . . , N ;
•eM = maxk {|e(k)|}, the maximal absolute value of the
tracking error, is used as index of measure of transient performance;
2
•L2 [u] = N1 N
k=1 |u(k)| , the average control input, is
used to evaluate the amount of control effort;
•C[u] = L2 [Δu]/L2 [u], the normalized control variations,
is used to evaluate the input chattering, where L2 [Δu] =
N
2
1
k=1 |Δu(k)| , is the average of control input increments;
N
M
2
1
•L2 [qess] = M
j=1 |qess(j)| , the average quasisteady state error specially defined to evaluate the point-to-point
trajectory (T1 or T2) tracking error when the reference trajectory
is a constant, i.e., qess(j) = e(k)|θd ≡C , is used as an objective
numerical measurement of quasi-steady state performance;
•qessM = maxj {|qess(j)|}, the maximal absolute value of
the quasi-steady state error, is specially defined to evaluate the
point-to-point trajectory (T1 or T2) tracking control and can be
used as index of measure of quasi-steady state performance.
C. Comparative Experimental Result
The results of the bending angle trajectory tracking are presented in Fig. 5 and the parameter estimations are shown in the
Appendix C. The performance indexes are given in Table IV. The
PID (C1) controllers perform relatively well in terms of L2 [qess]
due to the effect of integral item. However, conventional PID
controllers generally do not work well for nonlinear systems,
high order and time-delay systems, and particularly complex
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IEEE ROBOTICS AND AUTOMATION LETTERS, VOL. 5, NO. 4, OCTOBER 2020
Fig. 5. Experimental results of bending angle trajectory tracking control: (a)(d) are tracking results of trajectory T1, T2, T3 and T4, respectively; (e)-(h) are
corresponding tracking error of trajectory T1, T2, T3 and T4, respectively.
TABLE IV
PERFORMANCE INDEX
and vague systems that have no precise mathematical models
[35]. This is caused by the low controller order and constant
feedback gains. Thereby, PID controllers perform poorly in
terms of performance indexes L2 [e], eM and qessM .
By contrast, the SMC (C2) controllers present obviously better performances in terms of performance indexes L2 [e], eM and
qessM , while the average quasi-steady state errors are relatively
bigger in terms of L2 [qess]. The reason is that transient and
Fig. 6. Control mode and control input (duty cycle, %) of proposed ARC
(C3) controller: (a)-(d) are control mode of trajectory T1, T2, T3 and T4,
respectively; (e)-(h) are control input (duty cycle, %) of trajectory T1, T2, T3
and T4, respectively.
steady-state performances of smoothed SMC law are quantified
by Equation (32), thus the tracking error is bounded above by
a known function which exponentially converges to a specified accuracy (namely, a relatively larger steady state tracking
error).
Combining the smoothed SMC law and the parameter adaptation law, the resulting transient performances quantified by
Equation (32) suggest that dynamic tracking error is bounded
above when the error convergence quantified by Equation (33)
suggests that resulting tracking error can converge to zero.
Therefore, the ARC (C3) controllers achieve best performance
for all trajectories in terms of performance indexes L2 [e], eM ,
L2 [qess] and qessM when comparing with the controllers
above.
The control mode and control inputs of ARC (C3) controllers
are presented in Fig. 6. When tracking the same trajectory, all
controllers use the similar amount of control effort in terms
of L2 [u1 ] and L2 [u2 ], yet the degrees of input chattering are
different in terms of C[u1 ] and C[u2 ]. Specifically speaking,
the degrees of input chattering of PID (C1) controllers are
normally larger, the reason is that differential item is sensitive
to the measurement noises. By contrast, the degrees of input
chattering of smoothed SMC (C2) are normally smaller owing
to the effectiveness of boundary layer. For the same reason, the
degrees of input chattering of ARC (C3) controllers are usually
smallest or similar to SMC (C2) controllers resulting from the
normally smaller K and/or larger ε1 and ε2 .
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CHEN et al.: FIBER-REINFORCED SOFT BENDING ACTUATOR CONTROL UTILIZING ON/OFF VALVES
V. CONCLUSION
This study investigated the bending angle trajectory tracking
control of the FRSBA. Because a tractable theoretical dynamic
FRSBA model had been lacking, a second-order transfer function model with parametric uncertainties was first identified
using black-box system identification with a multi-sine excitation signal. Then, nonlinear pneumatic dynamics with model
compensation were established by considering the performance
degradation caused by the valves disregarded dynamic characteristics. Subsequently, an adaptive robust controller was designed to handle the systemic nonlinearities while ensuring satisfactory transient performance. In this process, the asymptotic
trajectory tracking with guaranteed transient and steady-state
performances in the presence of parameter uncertainties of the
proposed ARC controller was theoretically proven. Finally, the
validity of the system dynamic model and the effectiveness of
the proposed ARC controller were verified through comparative experiments. The experimental results and quantitative
performance indexes illustrated the high-performances of ARC
method.
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