1410 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997
Neural-Network-Based Robust Fault
Diagnosis in Robotic Systems
Arun T. Vemuri, Member, IEEE, and Marios M. Polycarpou, Member, IEEE
Abstract—Fault diagnosis plays an important role in the operation of modern robotic systems. A number of researchers have
proposed fault diagnosis architectures for robotic manipulators
using the model-based analytical redundancy approach. One of
the key issues in the design of such fault diagnosis schemes is
the effect of modeling uncertainties on their performance. This
paper investigates the problem of fault diagnosis in rigid-link
robotic manipulators with modeling uncertainties. A learning
architecture with sigmoidal neural networks is used to monitor
the robotic system for any off-nominal behavior due to faults. The
robustness and stability properties of the fault diagnosis scheme
are rigorously established. Simulation examples are presented
to illustrate the ability of the neural-network-based robust fault
diagnosis scheme to detect and accommodate faults in a two-link
robotic manipulator.
Index Terms— Adaptive law, analytical redundancy, fault accommodation, fault diagnosis, learning architecture, nonlinear
fault diagnosis, neural networks, robotic systems, robust fault
diagnosis.
I. INTRODUCTION
ROBOTIC systems are integral components of many com- plex engineering systems including manufacturing processes [1] and space-based systems [2]. Stricter operational
and productivity requirements in such systems are resulting in
robotic manipulators working near their design limits for much
of the time. This may often lead to robotic system failures
which are typically characterized by critical changes in the
robotic system parameters or even by nonlinear changes in the
inherent dynamics of the manipulator. Robotic system failures
can potentially result not only in the loss of productivity
but also can lead to unsafe operation of the system. In
general, modern control systems which are designed to handle
small perturbations that may arise under “normal” operating conditions (in the “linear” regime) cannot accommodate
abnormal behavior due to faults. Hence automated health
monitoring of robotic systems and effective accommodation
of any faults play a crucial role in the operation of modern
robotic systems and especially autonomous and intelligent
robotic manipulators.
The design and analysis of fault diagnosis (FD) architectures
for robotic systems using the model-based analytical redundancy approach has received considerable attention [2]–[4].
Manuscript received April 14, 1996; revised September 23, 1996, February
20, 1997, and August 17, 1997.
A. T. Vemuri is with the Department of Engine and Vehicle Research,
Southwest Research Institute, San Antonio, TX 78238-5166 USA.
M. M. Polycarpou is with the Department of Electrical and Computer
Engineering, University of Cincinnati, Cincinnati, OH 45221-0030 USA.
Publisher Item Identifier S 1045-9227(97)08093-4.
In this approach, quantitative nominal models of the robotic
system together with sensory measurements are used to provide estimates of measured and/or unmeasured variables. The
deviations between estimated and measured signals provide
a residual vector which can be utilized to detect and isolate
system failures. In general, a fault is declared if a measure of
the residual vector exceeds a certain threshold value. An alternative to analytical redundancy is the hardware redundancy
approach, where additional physical instrumentation is used to
provide the necessary redundancy [5].
The appeal of model-based FD schemes lies in the fact that
the redundancy required for detecting faults is created using
powerful information processing techniques without the need
of additional physical instrumentation in the system. However,
the model-based FD approach relies on the key assumption that
a mathematical characterization of the manipulator is known a
priori. In practice, this assumption is usually not valid since it
is difficult to obtain the necessary modeling accuracy required
for the construction of reliable analytical redundancy-based
FD architectures. Unavoidable modeling uncertainties, which
arise due to modeling errors, time variations, measurement
noise, and external disturbances, deteriorate the performance
of FD schemes by causing false alarms. This necessitates
the development of FD algorithms which have the ability
to detect manipulator failures in the presence of modeling
uncertainties. Such algorithms are referred to as robust fault
diagnosis schemes.
The construction of robust FD architectures for robotic
manipulators has been investigated to a limited extent. In
[6], Schneider and Frank use threshold adaptation based on
fuzzy logic to improve robustness of state-space model-based
FD architectures. Time-varying state-dependent thresholds are
used in [7] to achieve robustness in parity relations based
FD schemes for remote robots. These studies rely on two
key assumptions: 1) the nominal model of the system is
linear and 2) the failures are modeled as external additive
inputs (functions of time). Although it is convenient from an
analytical viewpoint to study the FD problem in a linear system
framework, the dynamics of robotic systems are inherently
nonlinear. Furthermore, most practical failures are nonlinear
functions of the state and input.
This paper presents a learning methodology for robust fault
diagnosis in rigid-link robotic manipulators, which is based on
a nonlinear nominal model of the manipulator and nonlinear
deviation faults. The modeling uncertainties are assumed to be
bounded while the faults are modeled as nonlinear functions
of the measured variables. The principal idea behind this
1045–9227/97$10.00 1997 IEEE
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VEMURI AND POLYCARPOU: ROBUST FAULT DIAGNOSIS IN ROBOTIC SYSTEMS 1411
approach is to monitor the plant for any off-nominal system
behavior (which could be either due to faults or uncertainties)
utilizing a sigmoidal neural network. By using the knowledge
of the bound on the uncertainty we develop a systematic
procedure, based on neuro-control techniques, for identifying
the effects of system failures in the presence of modeling
uncertainties. The neural network not only is used to detect
the occurrence of the fault but it also provides a postfault
model of the robotic manipulator. This postfault model can
be effectively used to isolate and identify the fault and, if
possible, for accommodation of the failure.
The fault diagnosis scheme described in this paper is rigorously analyzed for robustness and stability. Specifically, the
robustness result addresses the FD system performance in the
presence of modeling uncertainties prior to the occurrence of
any faults while the stability property characterizes the FD
system performance after the occurrence of the fault.
The organization of this paper is as follows: In Section II,
the robot dynamics and its control law are described, and
the fault diagnosis problem is formulated. In Section III,
the neural-network-based robust fault diagnosis scheme is
described. The analytical properties of the robust FD algorithm are established in Section IV. Simulation examples
illustrating the performance of the FD algorithm on a two-link
robotic manipulator with modeling uncertainties are presented
in Section V. Section VI has some concluding remarks.
II. PROBLEM FORMULATION
| Consider a robotic manipulator described by | model a much larger class of failures [in the framework of (4)] is the need to approximate unknown nonlinear functions. |
AssignmentTutorOnline
| (1) and software simulation tools have rendered possible the use |
|
| where are vectors of joint positions, velocities and accelerations, respectively, is the input torque of on-line approximators such as sigmoidal neural networks for constructing and analyzing nonlinear models [12], [13]. |
|
| vector, is the inertia matrix (whose inverse | In light of the above, the objective of this paper is to |
exists [8], [9]), is a matrix containing the
centripetal and Coriolis terms, is the gravity vector,
is a vector containing the unknown static and
dynamic friction terms, and is a vector representing
| unknown additive bounded disturbances and noise. The term is a vector which represents the fault in the robot manipulator, represents the time profile of the fault, and is the time of occurrence of the fault. The control objective of the robotic system (1) is to follow a desired trajectory. A number of techniques are available in the literature for deriving position control laws for robotic manipulators in the presence of modeling uncertainties and in the absence of faults (i.e., ) [8], [10], [11]. Without any loss of generality, in this paper we use the computed-torque method to obtain a trajectory-tracking controller for the robotic manipulator described by (1). The controller derived using this method relies on the position and velocity measurements of |
A1) The failure is abrupt and occurs at some unknown time i.e., the time-profile of the failure is given by |
|
| each link and the nominal model given by | (2) | III. FAULT DIAGNOSIS ARCHITECTURE In this section, we describe a robust nonlinear fault diagnosis |
| if | ||
| if | ||
| A2) The robotic system states remain bounded after the occurrence of a fault; i.e., . A3) The modeling uncertainty is bounded; i.e., |
||
| where is a known constant and is | ||
| some compact domain of interest. |
The structure of the computed torque controller is described by
(3)
where is the desired trajectory, is the tracking
error, is an diagonal matrix of damping gains,
is an diagonal matrix of position gains. If the robot
dynamics are known exactly, then these matrices can be chosen
so that the control law leads to an exponentially convergent
tracking error [8], [9].
In the remaining portion of the paper, since the above
computed torque control law is a function of and only,
we represent the robotic manipulator model (1) as
(4)
where . Note that the friction and the
disturbance terms in (4) are assumed to represent the modeling
uncertainties in the system.
A fault in the robotic system changes the dynamics of the
manipulator in an unpredictable way. An accurate description
of fault conditions, most often, requires nonlinear modeling of
faults, which is what is described by in (4). The nonlinear
modeling capability is reflected in allowing the deviation
due to faults to be a nonlinear function of the joint positions
and velocities. It is important to note that the fault formulation
described by (4) allows nonadditive types of faults. For
example, if the matrix changes to due to a fault
at time then this can be represented by letting
We refer to FD schemes that
are based on such nonlinearly modeled faults as nonlinear FD
schemes. The price that one has to pay for the potential to
However, recent advances in both hardware implementation
develop a robust nonlinear fault diagnosis architecture with
guaranteed robustness and stability properties for the robotic
system described by (4). We make the following assumptions
throughout the paper.
architecture for detecting system faults in robotic manipulators described by (4). We begin by observing that in the
absence of modeling uncertainties any off-nominal behavior
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1412 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997
observed from input–output measurements can be attributed
to a fault in the robotic system. Thus, the process of fault
detection in the absence of modeling uncertainties can be
achieved by approximating, on-line, the unknown function
[14]–[16]. However, in the presence of modeling
uncertainties, the difference in the dynamics could be either
due to faults or due to modeling uncertainties. Therefore, a
key question is: how does one identify the effects of a fault in
the presence of modeling uncertainties? In the special case that
the modeling uncertainties have certain known characteristics
that distinguish them from faults, then any off-nominal system
behavior provided by the neural network can be classified
(for example, via pattern and signal classification methods) as
being due to either faults or modeling uncertainties. However,
in most practical manipulators, both faults and modeling uncertainties are unknown a priori. Hence the issue of robustness
is very important.
Frank [17] describes a robust fault detection algorithm
which, while providing an on-line estimate of the uncertainty
level, deals exclusively with the detection of faults. In the context of fault diagnosis, in addition to detecting the occurrence
of a fault, it would be useful to obtain an approximation of
the fault function. In this paper, we develop a robust fault
diagnosis algorithm for detecting faults in the presence of
modeling uncertainties that satisfy the bounding condition A3.
An estimate of the fault function is obtained provided that the
ratio between the fault function magnitude and the modeling
uncertainty level is sufficiently large. Note that Assumption
A3 allows the derivation of a robust fault diagnosis algorithm
which is based on bounded unstructured uncertainty.
A. Nonlinear Estimation Model
A key objective of this paper is to design a fault diagnosis
architecture for the robotic manipulator described by (4) using
the approximation properties of sigmoidal neural networks. In
this section, the construction of a neural-network-based nonlinear estimation model is described. Utilizing this estimation
model, a learning algorithm for updating the parameters of the
neural network so that it approximates any off-nominal behavior due to faults, in the presence of modeling uncertainties that
satisfy Assumption A3, is described in the next section.
We consider an estimated model of the form
(5)
where is the estimate of the velocity vector of the
manipulator joints, is a design constant, is a threelayered sigmoidal neural network and represents the
adjustable weights of the network in vector form. If the number
of neurons in the hidden layer is , then (see the
Appendix for details of this representation of a three-layered
sigmoidal neural network).
The estimation model (5) is a nonlinear observer-type
scheme that can be implemented in the form of stable filter as
where is the output of the first-order filter ,
with the filter input given by
The construction of an appropriate estimation model, able
to follow any changes in the input–output behavior of the
physical system, is a crucial component in the development
of the overall fault detection scheme. The output of the above
nonlinear estimation model is used to update the weights of the
neural network. The nonlinear estimation model (5) is not only
easy to implement but, more importantly, has some desirable
stability and performance properties, which are presented in
the next section.
The initial weight vector, of the neural network
is chosen such that
(6)
corresponding to the no-failure situation, while the initial value
of the estimator is selected as . Note that the weight
initialization given by (6) can be achieved by simply setting the
weights of the output layer to zero. Starting from these initial
conditions, the main objective is to adjust (using input–output
information) the weight vector at each time so that
approximates the unknown function
Once this is achieved then the output of the neural network
can be used to detect, diagnose, and accommodate system
failures.
B. Learning Algorithm
Based on the estimation model (5), we now present a
learning algorithm for updating the adjustable parameters of
the sigmoidal neural network. We start by defining
as the error between the measured velocity vector and its
estimate. Then using (4) and (5) we obtain the following error
dynamics:
The above dynamics can be written as
(7)
where denotes the network approximation error [18], [19]
and is defined as
The network approximation error is a critical quantity,
representing the minimum possible deviation between the
unknown function and the neural network . Ideally we
would like to have ; in other words, we wish to
be able to approximate the off-nominal behavior of the system
due to faults exactly. Unfortunately, this is not always possible,
and hence the network approximation error needs to be
considered. Since is unknown, the approximation error
cannot be calculated a priori. Several factors including the
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VEMURI AND POLYCARPOU: ROBUST FAULT DIAGNOSIS IN ROBOTIC SYSTEMS 1413
number of hidden layers in the neural network and the number
of weights (denoted by influence the value of . Universal
approximator results for sigmoidal neural networks indicate
that if is sufficiently large then can be made arbitrarily
small on a compact region (assuming is continuous) [20].
The vector (an “artificial” quantity required only for
analytical purposes) is the value of that minimizes the
-norm between the fault function, , and the neural
network, over all in the compact learning
domain subject to the restriction that belongs to a
compact convex region ; i.e.,
In addition to the factors described above, the choice of
influences the approximation error and the weight vector
[15].
Under basic smoothness conditions on the error
dynamics (7) can be written as
(8)
where is the weight estimation error, is
the gradient of the neural network with respect to its adjustable
weights; i.e.,
and
represents the higher-order terms (with respect to of the Taylor series expansion. This higher-order term encapsulates the
nonlinear parameterization structure of the sigmoidal neural
network. Using the mean value theorem [21] it can be shown
that for satisfies
with
where Note
that for each we have i.e.,
as the weight estimates of the neural network, converges
to the optimal weight, the effect of diminishes. In
the special case of linearly parameterized networks, is
identically equal to zero.
Now if we define then the error equation (8)
can be written as
(9)
Since the initial condition for the error equation
is .
Based on the error dynamics described by (9), we propose
the following adaptive laws for updating the weights of the
neural network:
where is a positive definite learning rate
matrix and is the dead-zone operator, defined as
if
otherwise
where is an -dimensional vector of zeros and
(10)
The projection operator (which restricts the parameter
estimate vector to the compact, convex region is
used to avoid parameter drift, a phenomenon that may occur
with standard adaptive laws in the presence of modeling
uncertainties [22]. If is chosen to be a hypersphere of
radius , then the above adaptive law can be expressed as
(11)
where denotes the indicator function given by
if or
and
if and .
According to the error dynamics (9), the presence of modeling uncertainties cause a nonzero estimation error even in the
absence of a fault. The dead-zone term prevents the adaptation
of the neural-network weights when the estimation error is
within a (small) bound, thus enhancing robustness in the
fault detection scheme. The projection operator is required
in order to guarantee stability of the overall fault detection
scheme in the presence of network approximation errors. It
is noted that although a known bound on the modeling
uncertainty is required for selecting the size of the dead-zone,
the learning algorithm requires no knowledge of a bound on
the network approximation error or an upper bound on .
As expected, however, the actual value of does affect the
learning performance of the fault diagnosis scheme after the
occurrence of the fault (as shown in the next section).
IV. ANALYTICAL PROPERTIES
In this section we examine the robustness and stability
properties of the robust nonlinear FD scheme. The robustness
analysis deals with investigating the behavior of the neural
network in the presence of modeling uncertainties prior to
the occurrence of a fault. The stability analysis examines the
behavior of the neural network after the occurrence of a fault.
Theorem 4.1 (Robustness): The nonlinear fault detection
scheme described by (5), (11) guarantees
for prior to the occurrence of a fault.
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1414 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997
Proof: The error dynamics in the time period prior
to the occurrence of the fault are described by
(12)
Recalling that the initial parameter vector is chosen such
that . the primary question we need to examine
is whether the parameter estimates would start adapting, or
equivalently, whether , prior to the occurrence of a
fault.
Suppose (for the sake of contradiction) that there exists a
time (where such that
(13)
where denotes the first time instant that reaches the
dead-zone bound.
Using (11) and the continuity of , the parameter vector
has not been adapted in the interval . Furthermore,
by the continuity of we have ; in other words,
. Hence, from (9), the velocity estimation
error for is given by
Therefore
Now, since , this is clearly a
contradiction of (13), which implies that for the
estimation error remains within the dead-zone and the
output of the neural network remains zero.
The above theorem states that if there are no faults in
the dynamic system (i.e., ), then the output of the
neural network will remain zero (i.e., ) even in the
presence of modeling uncertainties. Equivalently, if the neuralnetwork output is nonzero then it is guaranteed that a fault has
occurred. (We note that a failure can also be declared whenever
.) Therefore, the neural network is robust with respect
to modeling uncertainties that satisfy Assumption A3.
We now examine the stability properties of the robust FD
algorithm. The stability analysis of the robust FD algorithm
deals with the boundedness of all the signals in the fault
detection system. This analysis also characterizes the bounds
of the velocity estimation error.
Theorem 4.2 (Stability): In presence of faults, the robust
nonlinear FD scheme described by (5) and (11) has the
following properties.
1) and are uniformly bounded.
2) There exists a nonnegative constant such that for any
finite
(14)
Proof: The stability properties of the error dynamics (9)
and the adaptive law (11) are analyzed using Lyapunov’s direct
method [22]. Consider the Lyapunov function candidate
if
if .
The time derivative of along the solution of (9) and (11)
when satisfies trivially. When , the time
derivative of satisfies
(15)
We next show that
(16)
Suppose that and ; if these conditions
are not satisfied then and the inequality holds trivially.
By completing the squares, the term is expressed as
Since by definition , we obtain that which
proves the inequality (16).
Using this inequality, (15) can be rewritten as
(17)
By combining the above inequality which guarantees that
for
and the fact that we establish that
Furthermore, by integrating (17) from to , we
obtain
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VEMURI AND POLYCARPOU: ROBUST FAULT DIAGNOSIS IN ROBOTIC SYSTEMS 1415
TABLE I
MANIPULATOR PARAMETERS
where
The above theorem guarantees the uniform boundedness of
the error signal and weights in the neural-network-based FD
scheme. Furthermore, (14) implies that the extended -norm
of the estimation error vector is, at most, of the same order
as the extended -norm of the modeling uncertainties (represented by )and the network approximation error (described
by ). Thus, this theorem quantifies the overall performance
of the learning scheme and the effect of the design parameter
on the performance.
Remark: The learning algorithm only uses the knowledge
of the size of the modeling uncertainties and does not require
an a priori known bound on the network reconstruction error.
Note that the projection operator in the weight update law deals
with the network approximation error while the dead-zone
operator is used to achieve robustness in the fault diagnosis
architecture.
V. SIMULATION RESULTS
This section illustrates the robust nonlinear FD scheme for
robotic manipulators via simulations. The robotic system used
for simulation purposes is modeled as two rigid links of lengths
and with point masses and at the distal ends of
the links. The dynamic equations of the robotic manipulator
are given in [8]. The link parameters of the manipulator are
given in Table I [23].
We assume that the robotic system has the following modeling uncertainties [23]:
The controller gains in the control law (3) are chosen as follows: where represents an
identity matrix of dimension . The position and velocity
of each joint is assumed to be available for measurement in
| order to be able to implement this controller. The domain of interest is |
The fault model provided by the neural network can be used for identifying the failure mode by comparing it with any known failure modes. Signatures of such known failure |
The units of and are rads and rads/s, respectively. The
bound on the modeling uncertainty in is . We use
a three-layer sigmoidal neural network with 35 neurons in the
hidden layer and two neurons in the output layer for detecting
system failures. The initial parameter vector of the network is
chosen such that the output of the neural network is zero in
(corresponding to the no-failure situation). This can be simply
achieved by setting the output weights of the neural network
to zero. The inputs to the neural network are the position and
velocity measurements of the robotic manipulator. We set the
learning rate as and the constant . The
dead-zone size is chosen as . The size of
the hypersphere for the projection algorithm is selected as
in this experiment.
We perform two sets of simulations. In the first simulation,
we consider a fault which occurs at and results in
a 75% change in the mass of link 1. This causes nonlinear
changes in the terms and of the robotic
system (4). In other words, the fault results in a nonlinear
change (which depends on the current positions and velocities
of the joint) in the dynamics of the robotic manipulator. In
the second simulation, we consider a fault that occurs due to
a tangle of complex factors; this fault is assumed to manifest
itself as a nonlinear change (in the robotic system dynamics)
described by
(18)
This fault is also assumed to occur at .
Fig. 1 shows the joint angle of each link, when the fault
results in a 75% change in the mass of link 1. It can be
seen that the robotic system tracks the desired trajectories
before the occurrence of the fault with a small tracking error,
which results due to the presence of modeling uncertainties.
Furthermore, it can be seen that the fault causes considerable
tracking error in the system for .
Fig. 2 shows the time histories of the fault functions and
the neural-network outputs. It can be seen that the outputs of
the neural networks remain zero prior to the fault and jump to
nonzero values soon after the occurrence of the fault. Thus
the neural-network outputs can be effectively used for the
detection of system faults in this simulation example.
The learning capability of the fault diagnosis architecture
is also illustrated in Fig. 2. This figure shows that the neuralnetwork outputs provide a good approximation of the fault
functions in each link. The presence of modeling uncertainties
and the network approximation error causes the observed
mismatch between the fault function and the neural-network
outputs. Thus the fault diagnosis scheme proposed in this
paper is able not only to detect the occurrence of a fault in
the presence of modeling uncertainties but it also provides a
rough model of the fault (described by in the robotic
manipulator.
modes of the relevant robotic manipulator can be stored in
a postfailure model bank. Pattern recognition techniques and
associative memories provide effective means for comparison
purposes. In general, it is important that the postfailure model
bank include (and be updated periodically) the signatures
of recurring and well-understood failures. In many cases,
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1416 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997
Fig. 1. Joint angles of the manipulator with fault that results in 75% change in mass of link 1.
Fig. 2. Time histories of the fault functions and the neural-network output when the fault results in 75% change in mass of link 1.
however, unexpected failure situations are encountered. This
may occur as a result of minimal knowledge of possible
manipulator faults, unexpected breakdowns, or even as a
result of errors in assembling the postfailure model bank. The
construction of a postfailure model bank and its effective use
for fault identification and fault isolation based on the postfault
models provided by the proposed fault diagnosis methodology
needs further investigation.
Fig. 3 shows the joint angle of each link when a fault
described by (18) occurs. It can be once again seen that
the tracking of the robotic system deteriorates considerably
after the occurrence of the fault (i.e., for s). Fig. 4
shows the plot of the neural-network output and the fault
function. From this figure, it can be inferred that the neural
network provides a sufficiently good approximation of the fault
function.
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VEMURI AND POLYCARPOU: ROBUST FAULT DIAGNOSIS IN ROBOTIC SYSTEMS 1417
Fig. 3. Joint angles of the manipulator when the fault occurs due to a tangle of complex factors.
Fig. 4. Time histories of the fault functions and the neural-network output when the fault occurs due to a complex tangle of factors.
The postfault model provided by the sigmoidal neural
network can also be used for failure accommodation purposes.
One of the nonlinear control tools available for controller
reconfiguration purpose is feedback linearization [24]. The
main idea behind feedback linearization is to transform the
nonlinear system into a system with linear dynamics through
a change of coordinates and nonlinear feedback. If feedback
linearization is achievable (see [24] for conditions under which
a system is feedback linearizable), then it is possible to
obtain first, cancellation of the nonlinear functions and second,
desired closed-loop performance through the application of
powerful linear control design methodologies. We employ this
technique in this paper to reconfigure the control law (3).
Based on the postfault robotic system model given by
and using the feedback linearization technique, the control law
(3) can be reconfigured to
where is the reconfigured control law.
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1418 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997
Fig. 5. Joint angles plots illustrating accommodation when the fault results in a 75% change in the mass of link 1.
Fig. 6. Joint angles plots illustrating accommodation when the fault occurs due to a tangle of complex factors.
Fig. 5 shows the trajectories of the robotic system when the
reconfigured control law is used in the first fault case which
results in 75% change in mass of link 1. By comparing Figs. 1
and 5, it can be concluded that the trajectory tracking error is
considerably reduced by using the reconfigured control law
(19).
The accommodation of the fault (18) is illustrated in Fig. 6.
Once again, by comparing Figs. 3 and 6 it can be seen that
tracking of the desired trajectories is considerably improved.
Thus the proposed fault diagnosis scheme helps in providing a
postfailure model that can be used for accommodating system
failures via control reconfiguration.
In the above simulation study we have considered a fault
that causes a 75% change in the mass of link 1 in order to
illustrate the ability of the accommodation scheme to handle
such large faults. Although such large faults are relatively
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VEMURI AND POLYCARPOU: ROBUST FAULT DIAGNOSIS IN ROBOTIC SYSTEMS 1419
easily to detect, the proposed fault diagnosis scheme can
detect smaller faults as long as the ratio between the fault
function magnitude and the size of the modeling uncertainties
is sufficiently large. Further simulation study indicated that
for the uncertainty level considered here, the fault diagnosis
scheme is able to detect changes of mass greater than or equal
to 10%. As the analysis in Section IV shows, the performance
of the fault diagnosis scheme can be enhanced by improving
the accuracy of the nominal model.
VI. C
.
ONCLUSIONS
A neural-network-based learning methodology with guaranteed robustness and stability properties is described for the
detection of faults in robotic systems with modeling uncertainties. The nonlinear modeling approach is one of the key
features of the algorithm proposed in this paper. This feature
enables the development of a postfault robotic system model
which can be used to isolate as well as accommodate robotic
system faults. Simulation examples are used to illustrate the
effectiveness of the algorithm in detecting and accommodating
faults in a two-link robotic manipulator.
APPENDIX
NEURAL-NETWORK REPRESENTATION
The input–output characteristics of a three-layered sigmoidal neural network are described by
(20)
where is the input to the network, is the
output of the network, is the weight matrix from the input
layer to the hidden layer and is the weight matrix from the
hidden layer to the output layer. If the number of neurons in
the hidden layer is , then the dimensions of and
are and , respectively. The nonlinear
operator is the standard sigmoidal function
such that the th component of is given
by .
For the convenience of analysis, we represent the input–output characteristics of the neural network described
by (20) as
where is a vector-valued function such that the th output
has the form
(21)
The weight vector is given by
where operator stacks the columns of a matrix to
form a vector. From the above representation, it can be inferred
that , where .
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information for robust robot control,” in Proc. 26th Conf. Decision
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[6] H. Schneider and P. M. Frank, “Fuzzy logic-based threshold adaption
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[7] M. L. Visinsky, J. R. Cavallaro, and I. D. Walker, “A dynamic fault
tolerance framework for remote robots,” IEEE Trans. Robot. Automat.,
vol. 11, pp. 477–490, 1995.
[8] F. L. Lewis, C. T. Abdallah, and D. M. Dawson, Control of Robot
Manipulators. New York: McMillan, 1993.
[9] M. W. Spong and M. Vidyasagar, Robot Dynamics and Control. New
York: Wiley, 1989.
[10] D. M. Dawson, M. M. Bridges, and Z. Qu, Nonlinear Control of Robotic
Systems for Environmental Waste and Restoration. Englewood Cliffs,
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Authorized licensed use limited to: WRIGHT STATE UNIVERSITY. Downloaded on December 28, 2008 at 09:49 from IEEE Xplore. Restrictions apply.
1420 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997
Arun T. Vemuri (S’88–M’97) was born in India on
November 12, 1969. He received the B.E. (Honors)
degree in electrical engineering from University of
Roorkee, India, in 1991, the M.E. (Honors) degree
in systems science and automation from Indian
Institute of Science in 1993, and the Ph.D. degree in
electrical engineering from University of Cincinnati,
OH in 1996.
After completing his graduate studies, he joined
the Department of Engine and Vehicle Research,
Southwest Research Institute, San Antonio, TX, as a
Research Engineer. His research interests include intelligent control and fault
diagnosis in dynamic systems. He is currently invloved in the design and
implementation of supervisory control and fault detection systems for engines
and automotive transmissions.
Marios M. Polycarpou (S’87–M’93) was born in
Cyprus on August 27, 1962. He received the B.A.
(Cum Laude) degree in computer science and the
B.Sc. (Cum Laude) degree in electrical engineering
from Rice University, Houston, TX, in 1987, and
the M.S. and Ph.D. degrees in electrical engineering
from the University of Southern California, Los
Angeles, in 1989 and 1992, respectively. During
his undergraduate studies at Rice University from
1983 to 1987, he was the recipient of a Fulbright
scholarship.
In 1992, he joined the Department of Electrical and Computer Engineering
and Computer Science, University of Cincinnati, OH, where he is currently an
Assistant Professor. He teaches and conducts research in the areas of systems
and control, adaptive and intelligent control, neural-network learning, and
fault diagnosis.
Dr. Polycarpou is a member of the IEEE Steering Committee on Intelligent
Control. He was the Publicity Chair of the 1996 IEEE International Symposium on Intelligent Control and is currently serving as an Associate Editor on
the Conference Editorial Board of the IEEE Control Systems Society.
Authorized licensed use limited to: WRIGHT STATE UNIVERSITY. Downloaded on December 28, 2008 at 09:49 from IEEE Xplore. Restrictions apply.
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