This paper considers a decentralized multiple faults detection and isolation (FDI) scheme for reconfigurable manipulators. Inspired by their modularization property, a global sliding mode (GSM) based stable adaptive fuzzy decentralized controller is investigated for the system in fault free, while for the system suffering from multiple faults (actuator fault and sensor fault), the decentralized sliding mode observer (DSMO) is employed to detect their occurrence. Hereafter, the time and location of faults can be determined by a fault isolation scheme via a bank of DSMOs. Finally, the effectiveness of the proposed schemes in controlling, detecting and isolating faults is illustrated by the simulations of two 3DOF reconfigurable manipulators with different configurations successfully.
1. Introduction
Reconfigurable manipulators
[1]
, which are composed of a set of modules in standard size and interface, can be assembled into different configurations and geometries to undertake a variety of unknown or changeable tasks, such as the space exploration, smart manufacturing, high risk operations, and battle fields. However, working for a long time in the environments human can not be involved directly, faults will inevitably occur in actuator, sensor or other components, which may lead the control system to unstable or even destroyed. Therefore, the safety and reliability of reconfigurable manipulators becomes an urgent demand, and the efforts on its FDI are essential.
As a nonlinear and strong interconnected system, the controller design for reconfigurable manipulator is a quite active research field, and different approaches have been proposed during the past decades. The main challenging work lies in the dynamic parameter uncertainties caused by reconfiguration, friction, varying payloads and interconnection among joints. To against them, several schemes were developed for reconfigurable manipulators
[2

4]
, but they need to be redesigned when adds, removes or changes modules
[5]
. In contrast, the modular distributed control technology in
[6]
was more feasible. However, the communication time delay among the modules will lead to the inaccurate performance
[7]
. Take the modularization property and interconnection into account, decentralized control strategy was established
[8

12]
to overcome these shortcomings. It can not only effectively reduce the computational burden in centralized control, but also avoid communication time delay in distributed control, thus it is more feasible to implement to reconfigurable manipulators.
On the other hand, the investigated FDI approaches
[13]
can be essentially grouped into three main categories. Modelbased FDI is a powerful solution
[14]
. Based on the estimation error obtained from the strain gauge sensor signal and the reaction force observer
[15]
, the sensor fault can be detected, and then the faulty sensor signal was exchanged by the estimated one to maintain the stable control performance. In the presence of parametric uncertainties and actuator faults, a robust reliable control
[16]
for a near space vehicle was studied based on the fuzzy statespace observer. Based on signal processing technologies, a three latent space transformation based principal component analysis
[17]
has been effectively applied to detect the fault in the industrial processes with a large number of high correlated variables. In
[18]
, an artificial immunization algorithm was used to optimize the parameters in support vectors machine, it can avoid the premature convergence and guarantee the variety of solutions. Knowledgebased FDI is another effective method. A back propagation neural network (NN) was presented
[19]
to synthesize fault detection components based on the data collected in the training of autonomous robots. By virtue of soft computing techniques, an integrator of NN and fuzzy logic was proposed
[20]
for FDI of robot manipulators.
For reconfigurable manipulator systems, only several FDI schemes have been carried out. Inspired by the relationship between power efficiency and operation health conditions, a power efficiency estimation based health monitoring and fault detection technique was developed for modular and reconfigurable robots (MRR) with a joint torque sensor
[21]
. By the filtered torque estimation,
[22]
proposed a distributed fault detection scheme with torque sensing for MRR. In
[23]
, a hybrid controller has been constructed by a fault estimator and an impulse controller. Zhao et al.
[24]
compared the user defined threshold to the residual between the actual velocity and the observed one, to detect the fault occurrence or not, and the unknown input state observer was exploited for fault identification.
Sliding mode observer (SMO)
[25

28]
is a good idea for FDI. In
[29]
, SMO was employed to overcome the limitation that linear observer schemes cannot be utilized for sensor fault reconstruction. The authors in
[30]
performed a modular fault detection scheme, which did not require motion states of any other modules. In
[31]
, the sensor fault was converted into pseudoactuator fault scenario by a filter, and a decentralized sliding mode observer was recruited for the position sensor or velocity sensor fault identification.
This presented idea develops an adaptive fuzzy decentralized controller based on GSM for reconfigurable manipulators. For the purpose of FDI, the DSMO is employed to detect and isolate multiple faults, which can be judged when the observation errors beyond the predefined threshold. Hereafter, the time and location of fault occurrence can be decided by the fault isolation scheme. The main contributions of this work are as follows. (i) According to their modularization property, the decentralized controller is more feasible to reconfigurable manipulators, it implies that it is no need to redesign the controller when adds or removes the modules. (ii) The interconnection term is compensated to improve the joint tracking accuracy. (iii) The actuator and sensor fault can be detected precisely even they occur simultaneously. (iv) The fault time and location can be decided by isolating the multiple faults successfully.
The remainder of this paper is organized as follows. In Section 2, the dynamic models of reconfigurable manipulators with/without faults and the control objective are described. And then the decentralized controller is designed in Section 3. Subsequently, the FDI scheme is given in detail in Section 4. Hereafter, the effectiveness of the proposed methods is confirmed by the simulations in Section 5. Finally, the conclusions are drawn in Section 6.
2. Problem Statement
The dynamic model of
n
DOF reconfigurable manipulator with joint dynamic friction obtained by NewtonEuler formulation should be described as
where
q
∈
R^{n}
is the vector of joint displacements,
M
(
q
) ∈
R
^{n×n}
is the positive definite inertia matrix,
is the Coriolis and centripetal force,
G
(
q
) ∈
R^{n}
is the gravity term,
is the vector of joint dynamic friction torques, and
u
∈
R^{n}
is the applied joint torque.
For the development of decentralized control, each joint is considered as a subsystem of the entire manipulator system interconnected by coupling torques. By separating terms only depending on local variables
from those terms of other joint variables, each subsystem dynamical model can be formulated in joint space as
where
and
u_{i}
are the
i
th element of the vectors
and
u
, respectively.
M_{ij}
(
q
) and
are the
ij
th element of the matrices
M
(
q
) and
, respectively.
Property 1:
For scalar functions
M_{i}
(
q_{i}
) and
, the following condition holds
[9]
Let
, (2) can be expressed as the following state space equation
where
x_{i}
and
y_{i}
are the state vector and output vector of subsystem
S_{i}
, respectively. And
The dynamic model of the
i
th subsystem exhibits both actuator and sensor faults should be described by
where
α
(
t
−
T_{ta}
) and
α
(
t
−
T_{is}
) are step functions,
T_{ia}
and
T_{is}
are the occurrence times of actuator and sensor fault of the
i
th subsystem, respectively
and
are the actuator and sensor fault functions, respectively,
and
are the actuator and sensor fault terms, respectively.
Assumption 1:
The actuator and sensor fault terms
and
are unknown but norm bounded with
and
, where
δ_{i}
and
ρ_{i}
are positive constants.
The objective of this work is to detect and isolate the multiple faults occur in the
i
th subsystem in realtime based on the local joint information. In other words, fault detection is to determine the fault occurrence time, and fault isolation decides the fault location.
3. Decentralized Controller Design
The traditional sliding mode surface exists approaching mode and sliding mode during the whole sliding process, while the robustness of parameter uncertainties and external disturbances only present during the sliding mode. Yet the global sliding surface
[32

33]
can drive the system state to the sliding surface at the very beginning, the reaching interval can be eliminated.
In this section, an adaptive fuzzy controller only depends on the local joint information is designed to handle the joint tracking control of reconfigurable manipulators in fault free.
Define the tracking error
e_{i}
as
Then define the global dynamic sliding mode surface as
where
λ_{i}
> 0 ,
f_{i}
(
t
) is the designed function for reaching the global sliding mode, which satisfies the following three conditions:
(i) the initial
, where
e
_{i0}
and
are the initial values of
e_{i}
and
, respectively.
(ii)
(iii)
is existence.
According to the above conditions,
f_{i}
(
t
) should be
Yields,
Assumption 2:
The desired trajectories
q_{id}
are twice differentiable and bounded as
where
q_{iA}
is a known positive scalar.
Assumption 3:
The interconnection term
is explicitly bounded by
[24]
where
d_{ij}
≥ 0 ,
S_{j}
= 1+ 
s_{j}
 + 
s_{j}

^{2}
.
Define
, then substituting the time derivative of (13) into (2), one can obtain that
where
ϕ_{i}
(
x_{i}
) is the defined nonlinear function defined as
.
Now, the fuzzy logic systems are adopted to approximate the bounded nonlinear functions
ϕ_{i}
(
x_{i}
) ,
M_{i}
(
q_{i}
) and
, yields
where
ε
_{i1}
,
ε
_{i2}
and
ε
_{i3}
are the fuzzy logic system approximate errors, and
θ_{iϕ}
,
θ_{iM}
,
θ_{iC}
is the optimal parameter vectors satisfy the following conditions, respectively.
where Ω
_{iϕ}
, Ω
_{iM}
, Ω
_{iC}
,
U_{iϕ}
,
U_{iM}
and
U_{iC}
are the constraint set of
and
, respectively.
and
are the estimations of
ϕ_{i}
(
x_{i}
),
M_{i}
(
q_{i}
) and
that could be expressed as
where
and
are adjustable parameters to estimate
θ_{iϕ}
,
θ_{iM}
and
θ_{iC}
, respectively.
Define the minimum approximate error as
Assumption 4:
The approximate error of fuzzy logic system satisfies 
ε_{i}
 ≤
ε
_{0}
, where
ε
_{0}
> 0.
The GSM based adaptive fuzzy decentralized control law is designed as
where
k_{i}
is positive constant,
is the estimated value of
, then the estimation errors are defined as
where
and
are updated by
Theorem 1:
Consider the subsystem dynamic model (2) with assumptions 24, utilize the control law (27)(29) and adaptive laws (34)(37), all the variables of the closedloop system are guaranteed to be bounded, and the
H
_{∞}
tracking performance is satisfied.
Proof:
Choose the Lyapunov function candidate, i.e.,
Take property 1 into account, the time derivative of (38) is
Substituting (17)(19) and (28) into (39), we have
It’s worth noting that 
s_{i}
 ≤ 
s_{j}
 ⇔ 
S_{i}
 ≤ 
S_{j}
 , and by virtue of Chebyshev inequality,
Substituting (29), (34)(36) and (41) into (40), we have
Then substituting (37) into (42),
Integrating Φ(
t
) from 0 to
t
yields
Therefore
. Since
V
(0) is positive and finite,
exists and finite. According to Barbalat Lemma,
, it implies that
, namely, the tracking error
e_{i}
=
q_{id}
−
q_{i}
is convergent to zero asymptotically.
4. Fault Detection and Isolation
The aim to fault detection is not only to detect whether a fault occurs or not, but also to determine the time and location of fault occurrence, on which the decision can be made via a significant residual.
 4.1. Fault detection scheme
Define the output of the first order filter
z_{i}
as a new state variable
where
y_{i}
is the sensor output, and
a_{i}
>
b_{i}
≠ 0.
Substituting the output of (7) into (45), we have
Now, let
, the new state space expression of the subsystem is
where
It is obvious that
is controllable,
is observable.
Assumption 5:
There exists a proper matrix
L_{i}
satisfies
with the following Riccati function holds since
is observable
where
P_{i}
,
I_{i}
are symmetric positive matrices.
Assumption 6:
There exists arbitrary matrices
P_{i}
and
F_{i}
such that
where
F_{i}
= [
F
_{i1}
F
_{i2}
] ∈
R
^{1×2}
Define
and
are the estimated values of
and
, respectively, the state error
, output error
. The decentralized sliding mode observer (DSMO) based on a radial basis function neural network (RBFNN) can be established as
where
is the robust term to compensate the approximate error.
The observation error dynamic model derived from (47) and (50) is
where
is employed to compensate the interconnection term among the subsystems, where
,
and
are the estimation of
θ_{ip}
and
σ_{ip}
,
and
are their estimation errors, respectively.
θ_{if}
and
θ_{ig}
are the weights of ideal NNs, respectively.
σ
_{i(⋅)}
(⋅) is the NN basic function, and the approximate errors
ε_{if}
and
ε_{ig}
are norm bounded.
Using the RBFNNs to approximate the uncertainty term
and
g_{i}
(
q_{i}
) , where the ideal NNs as
and
denote the estimation of
θ_{if}
and
θ_{ig}
, the estimated errors are
and
, respectively. Then
Define the approximate error of the NNs as
where
.
Assumption 7:
The interconnection term
is bounded as
where
d_{ij}
> 0 is unknown constant, and define
.
Assumption 8:
The approximate error of the NN
ω_{i}
is bounded as 
ω_{i}
 ≤
β_{i}
with
β_{i}
is a positive scalar.
The NN weights
,
and
are updated by
Theorem 2:
Consider the subsystem dynamic model (2) with the assumptions 58, the observation error (51) obtained by the designed DSMO (50) and the subsystem state space equation (47), as well as the adaptive update laws (63)(65), the DSMO is asymptotically stable, it implies that
could converge to zero asymptotically.
The proof is similar to section 3, so it is omitted here.
From (51), we can know that if a fault occurs,
, and definite change will occur in
. Thus,
reflects the occurrence of faults. Summarily, the fault detection scheme can be devised as follows.
Fault detection scheme:
Faults can be detected if the residual
exceeds a predefined threshold
φ_{i}
. Otherwise, the system is faultfree within the considered time. The detection time
t_{id}
is defined as the first moment such that
is greater than
φ_{i}
.
 4.2. Fault isolation scheme
After a fault being detected, to determine its location is next objective, which is referred to fault isolation. In this subsection, based on the DSMO, we present a fault isolation scheme, which could avoid the affection caused by the faulty system to the faultfree subsystems.
Let
e_{ik}
(
k
=
s
,
a
) describes the observation error.
k
=
s
denotes the sensor fault. Define state observe error and the output error as
and
, the designed sensor fault observer is expressed as
where
where
ρ_{is}
is an adjustable constant,
ς
is a sufficiently small positive scalar, thereby the output error is limited within a neighborhood.
Assumption 9:
There exists arbitrary matrices
and
F_{is}
such that
where
F_{is}
= [
F
_{is1}
F
_{is2}
] ∈
R
^{1×2}
Comparing (47) to (66), the observation error dynamic could be expressed as
where
is utilized to compensate the interconnection, the RBFNNs
and
approximate the unknown terms
and
g_{i}
(
q_{i}
),
is a robust term, which could offset the approximation error.
k
=
a
denotes the actuator fault. Define the state observation error as
, output error as
, the designed actuator fault observer is described as
where
where
γ_{i}
is an adjustable scalar,
ς
is still a sufficiently small positive constant.
Comparing (47) to (70), the dynamic observation error is
Theorem 3:
Consider a bank of observers (66) and (70), the observation error dynamic model as (69) and (72), and the assumptions 59 with the adaptive laws (63)(65), the observation errors can converge to zero. In other words, the estimated state can follow to the actual state
. Once the subsystem is faulty,
e_{ik}
could definitely change.
Proof:
When
k
=
s
, choose Lyapunov candidate as
Its time derivative
Substituting (47) and (66) into (74), one can obtain that
According to similar proof in section 2, we have
It is clear that
V
(
t
) ≤
V
(0) , according to Barbalat Lemma,
, which means
e_{is}
converge to zero.
When
k
=
a
, choose the Lyapunov candidate function as
where
① The time derivative of
V
_{a3}
is
if
K
_{i3}
and
γ_{i}
satisfy (
a_{i}
+
K
_{i3}
) 
e
_{ia3}
 −
b_{i}

e
_{ia1}
 > 0 and
γ_{i}
−
α
_{i2}
> 0 at the same time. According to the sliding mode equivalent principle,
where (
γ_{i}
−
α
_{i2}
)sgn(
e
_{ia3}
) is the discontinuous equivalent output injection value.
② The time derivative of
V
_{a1}
is
if the selected
K
_{i1}
satisfies
K
_{i1}

e
_{ia1}
 − 
e
_{ia2}
 > 0.
③ The time derivative of
V
_{a2}
is
Noticing that 
e
_{ia2}
 ≤ 
e
_{ja2}
 ⇔
E_{i}
≤
E_{j}
, by virtue of Chebyshev inequality,
Substituting (85) into (84), we have
if the selected
K
_{i2}
satisfies
K
_{i2}

e
_{ia2}
 − 
ω_{i}
 > 0.
From the analysis above, it is clear
V
(
t
) ≤
V
(0), thus
e_{ia}
is bounded. According to Barbalat Lemma,
which means
e_{ia}
can converge to zero.
5. Numerical simulation
In order to verify the effectiveness of the proposed methods, two 3DOF reconfigurable manipulators with different configurations shown as
Fig. 1
are adopted for numerical simulation.
Reconfigurable manipulators for simulation
The friction terms of configuration
a
and configuration
b
are respectively expressed as follows:
Configuration a:
Configuration b:
The joint desired trajectories of configuration
a
are
The joint desired trajectories of configuration
b
are
The initial positions and velocities of each joint are
q
_{1}
(0) =
q
_{1}
(0) =
q
_{3}
(0) = 1 and
.
 5.1. Simulation for decentralized control scheme
In this subsection, the proposed decentralized controller (27) is applied to the reconfigurable manipulator in faultfree. The initial values of
and
are zeros. The control parameters are
η_{iϕ}
=
η_{iδ}
= 0.02 ,
ε
_{0}
= 0.01 ,
η_{iM}
=
η_{iC}
= 0.01 ,
k_{i}
= 5 ,
λ_{i}
= 25 ,
f_{i}
(
t
) =
f_{i}
(0)
e
^{−20t}
with
.
From
Fig. 2
, one can see that the actual trajectories take about only 0.2
s
to follow their desired trajectories, which means each joint in configuration
a
obtained good tracking performance.
Joint tracking curves of configuration a
To further test the effectiveness of the proposed decentralized controller, configuration
b
is employed for simulation with the same control law, and the
Fig. 3
shows excellent tracking performance as well. It demonstrates that the proposed control scheme is feasible to different configurations without any control parameters modification.
Joint tracking curves of configuration b
 5.2. Simulation for FDI
 5.2.1 Simulation results of fault detection
In this subsection, the predefined threshold of observation error is
φ_{i}
= 0.03 , the observer gains are
L
_{i1}
= 2000 ,
L
_{i1}
= 1000 and
L
_{i3}
= 30 , adaptive gains are
η_{if}
= 0.02 ,
η_{ig}
= 0.01 and
η_{ip}
= 0.01 , the robust coefficient is
β_{i}
= 0.01 , the filter coefficients are set as
a_{i}
= 1,
b_{i}
= 1 , respectively.
For the ith subsystem of reconfiguration manipulators, the RBFNNs are utilized to approximate the terms
and
. Where
X
= [
x
_{1}
, ⋯,
x_{n}
]
^{T}
is the input vector of NN,
σ
= [
σ
_{1}
, ⋯,
σ_{m}
]
^{T}
is the radial basis function, and the NN is expressed as
where
j
= 1, 2, …,
m
denotes the number of nodes in the hidden layer,
c_{j}
and
b_{j}
are the central and the width of RBFNN’s
j
th node,
where
i
= 1, 2,…,
n
denotes the number of hidden layers in NN, and
.
The injected faults are defined as
The observation error curves of configuration
a
and
b
shown as
Fig. 4
and
Fig. 5
, respectively, the red lines present the given thresholds. In these figures, from t=3s and t=5s, the actuator fault
f
_{1a}
and sensor fault
f
_{2s}
are embedded on joint 1 and joint 2, respectively. One can observe that the observation errors exceed the thresholds once the fault occurs. Therefore, this proposed scheme can detect the fault in real time.
Observation performance of configuration a in faulty
Observation performance of configuration b in faulty
 5.2.2 Simulation results of fault isolation
In this subsection, utilize DSMOs (66) and (70), the actuator fault observer gains are set as
k
_{i1}
=100,
k
_{i2}
=300,
k
_{i3}
=20. The adjustable sliding mode parameters are
ρ_{i}
= 130 ,
γ_{i}
=800 , the sensor observer parameters and the fault are as in section 5.2.1.
Fig. 6(a)
shows the actuator observer output curves, one can see that the observation error of joint 1 of configuration
a
exceeds the predefined threshold after t=3s, and that of joint 2 lies within the given threshold, it proofs that the actuator fault can be isolated from sensor fault. Meanwhile,
Fig. 6(b)
shows that of sensor fault, the same conclusion can be obtained. To further test the proposed isolation scheme, the configuration
b
is employed for simulation, whose results shown as
Fig. 7
. It’s worth noting that joint 3 is fault free according to the results of fault detection in the previous subsection, therefore, its simulation result can be omitted here.
The bank of observers’ output error curves of configuration a
The bank of observers’ output error curves of configuration b
6. Conclusions
By virtue of DSMO, a multiple faults detection and isolation schemes for reconfigurable manipulator are presented in this paper. Based on the modularization property and Lyapunov stability theory, the adaptive fuzzy decentralized controller is applied to the reconfigurable manipulator in faultfree. Then the multiple faults including actuator fault and sensor fault can be detected by means of the observation errors whether exceed the predefined thresholds or not. After being detected, multiple faults are isolated to decide which joint and which component is faulty. Its effectiveness has been demonstrated by the simulation of two 3DOF reconfigurable manipulators with different configurations in the presence of multiple faults.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (61374051) and the Scientific and Technological Development Plan Project in Jilin Province of China (20150520112JH).
BIO
Bo Zhao He received Ph.D. degree in the Department of Control Science and Engineering from Jilin University, China in 2014. Now, he is a postdoctoral at Institute of Automation, Chinese Academy of Science, China. His interest covers fault diagnosis and fault tolerant control, and intelligent control.
Chenghao Li He received M.E. degree in the Department of Control Science and Engineering from Jilin University, China in 2013. Now, he is a production engineer of FAW Car Co., Ltd. China. His interest covers fault diagnosis and fault tolerant control.
Tianhao Ma He received his M.E. degree in the Department of Control Science and Engineering from Changchun University of Technology, China in 2015. His interest covers intelligent control and robust control.
Yuanchun Li He received Ph.D. degree in the Department of General Mechanics from Harbin Institute of Technology, China in 1990. Now, he is a professor in the Department of Control Science and Engineering, Changchun University of Technology. His research interest covers complex system modeling and robot control.
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