Sensor and Actuator Fault Diagnosis Based on Soft Computing Techniques

2.1 The Manipulator Robot Model

The dynamic model of the robotic system can be derived via either the Lagrangian or Newton–Euler methods [10, 12, 18, 28]. Generally, a healthy manipulator arm is described by the following system of differential equations:

where θ,θ˙,θ¨ ∈ Rn denote the vectors of joint positions, velocities, and accelerations, respectively; τ ∈ Rⁿ is the vector of input torques; G(θ) ∈ Rⁿ is the vector of gravitational torque; V(θ,θ˙) ∈ Rn is the vector representing Coriolis and centripetal forces; M(θ) ∈ R^n×n is the inertia matrix whose inverse exists; and μ(θ,θ˙,τ,t) ∈ Rn denotes the unmodeled dynamics. It is assumed that the unmodeled dynamics are bounded.

2.2 Artificial Neural Network Principles and Application

The neural network is a technique that has the capacity of modeling a complex system. It is considered a “black box” model.

In this case, the operator does not need the physical theory because the neural network could learn the relationship between the input and the output. To get the desired outputs, the weights of artificial neural networks (ANNs) need to be adjusted using a learning algorithm such as the Levenberg–Marquardt algorithm. This training algorithm is more powerful and faster than the conventional gradient descent technique.

2.3 Fuzzy Logic Principles and Applications

The fuzzy logic control process consists of three different stages: fuzzification, control rules evaluation, and defuzzification. During the fuzzification stage, crisp values of the input variables are converted through the membership functions (MFs) and fuzzy sets into fuzzy values. Then, the fuzzified input values are evaluated through the control IF–THEN rules and the control output is generated. Fuzzy output is then converted back to crisp values through one of the various defuzzification methods [27].

2.4 Diagnosis Principles

The FDI scheme is divided into two steps: detection step and isolation step. The first is based on generating a signal called residual. It used like an indicator of the occurrence of the fault. These signals are the comparison between the real measurements of the output system (provided by the robot model) and those of the neural network (which indicate the normal operation). If the difference is equal to zero, then the equipment systems operate in a healthy state; otherwise (residual ≠ 0), they are affected by a fault, which should be isolated. The role of the second step is to identify the element attached by the fault. To get the adequate decision, this phase requires a technique that deals with uncertain data. The most appropriate technique is the fuzzy logic. It needs an expert system. Figure 1 shows the different stages of the adopted strategy.

Figure 1

Residual Generation and Neuro-fuzzy Decision.

2.5 Sliding-Mode Observer

The basic idea of the proposed approach for residual generation is the synthesis of a robust sliding-mode observer. Thus, the estimated states are used to generate robust residuals that can detect the presence of defects [6, 13, 31]. Figure 2 illustrates the principle of this method.

Figure 2

Principle of the Method based on the Sliding-mode Observer, Proposed for Residual Generation.

3.1 Residual Generation

In this article, a multilayer perceptron (MLP) with a back-propagation algorithm is used to reproduce the behavior of non-linear dynamical systems and a second MLP is used for residual classification. For a p-dimensional input vector and a q-dimensional output vector, the MLP input/output relationship defines a mapping from a p-dimensional Euclidean space to a q-dimensional Euclidean output space. Using only one hidden layer, presenting in the n-th sample (where n = 1, 2, …, n_p), the input vector X(n)=[X₁(n), X₂(n), …, X_p(n)]^T, the activation of the output neuron k (where k = 1, 2, …, q) is

where m is the number of neurons in the hidden layer, W_jk is the weight between the j-th neuron of the hidden layer and the k-th neuron of the output layer, W_ji is the weight between the i-th neuron of the input layer and the j-th neuron of the hidden layer, ϕ_k is the non-linear activation function of the output layer, and ϕ_j is the non-linear activation function of the hidden layer. In this article, the weights of the MLP are trained by the well-known back-propagation algorithm.

In discrete time, the state equation of a fault-free non-linear dynamic system is given by

where x(t) is the state vector at time t, u(t) is the applied control vector, Δt is the sample rate, and f(·) is the vector-valued non-linear function of the fault-free system.

Considering now that a fault i occurs, the dynamics of the system are modified to

where g_i(·) is the vector-valued non-linear function of the system affected by fault i. The faults may or may not be additive inputs (i.e., dependent only on the time variable). The i-th fault vector can be defined as the difference between the faulty system dynamics [Eq. (4)] and the fault-free system dynamics [Eq. (3)]:

Obviously, for the fault-free system, Ψ_i(t+Δt).

Generally, for each possible fault i, the fault vector Ψ_i has a particular behavior, called the fault signature. For the identification of the fault type, the fault vector must be computed and analyzed. Therefore, the dynamic behavior of the fault-free system [Eq. (3)] must be estimated (e.g., by the mathematical model or by an ANN). Then, when fault i occurs, the residual vector can be computed as

where f^ is a vector that represents the input/output mapping of the estimated fault-free dynamic behavior of the system and e_i is the error between the actual fault-free behavior and the estimated one. In real systems and the fault-free case, the error is due to external disturbances, unmodeled system uncertainty, mapping errors (or modeling errors in model-based systems), and measurement noise. In this work, an MLP is employed to reproduce (estimate) the fault-free dynamic behavior of a robotic manipulator. Taking as inputs the control signals u and the states x measured at t, the MLP reproduces the states x of the fault-free system measured at t+Δt [Eq. (7)].

For residual generation in mechanical manipulators and as mentioned above, an MLP is used to approximate the state equation (dynamic function) of the fault-free manipulator. The dynamic of a fault-free robotic manipulator with actuators in each joint is given by

where θ is the vector of joint angular positions, τ is the vector of joint torques, M is the inertia matrix, t is the time index, C is the vector of Coriolis and centrifugal forces, G is the vector of gravitational torques, F is the vector of frictional torques and other non-linearities, and τ_d is the vector of external uncorrelated disturbances. In this work, residual analysis for fault isolation purposes is performed with a neuro-fuzzy network utilizing the positions residuals. A general schema is shown in Figure 1.

Outputs 1 through q-1 correspond to the q-1 possible failure modes, while output q corresponds to fault-free operation. The ANN’s output i (i = 1, …, q-1) is trained to present a “1” in case fault i occurs and “0” otherwise. The output q is trained to present a “1” in case of a fault-free operation and “0” otherwise.

In the FDI procedure, the normalized positions applied at t are presented to the MLP, which estimates the positions at t+Δt; the MLP outputs are compared with the normalized positions and velocities measured at t + Δt to generate the residuals; the position residuals are presented to the neuro-fuzzy network, which classifies the residuals and generates a vector that indicates the operation status of the system.

3.2 Residual Evaluation

The task of residual evaluation can be achieved by a neuro-fuzzy decision. A neuro-fuzzy network is based on the association of fuzzy logic inference and the learning ability of neural networks. The neuro-fuzzy approach is a powerful tool for solving important problems encountered in the design of fuzzy systems, such as determining and learning MFs, determining fuzzy rules, and adapting to the system environment.

The main points of the residual evaluation procedure are described below.

3.2.2 Neural Network Structure

For fault diagnosis, it is desirable to use a neural network to model the non-linear relationship between the fuzzified residuals and the fault decision functions. An MLP network is therefore a good candidate. Moreover, to account for memory in the decision process, it is necessary to use a recurrent neural network (RNN). The RNN may be implemented as a neural model described by

(8)Dk(fi) = gi(φ(k)), (8)

where D_k(f_i), i = 1, …, n_f, are the fault decision functions also referred to as fault indicators and f_i are the faults acting on the process. The regression vector contains the fuzzy residuals R_i(k), i = 1, …, n_r, and the delayed decisions D_k(f_i), i = 1, …, n_f. Because of the feedback introduced, the recurrent neural model may be realized by a three-layer MLP. This is illustrated by the example given in Figure 3, which shows a residual evaluation scheme processing three residuals (r₁, r₂, r₃) to diagnose three faults (f₁, f₂, f₃). The corresponding neural network has the following architecture: an input layer with 12 units representing all possible states of the fuzzy residuals together with the past decisions, a hidden layer having 4 units, and an output layer with 3 units each assigned to a decision function. The use of this RNN architecture ensures reliable dynamic decision making [3].

Figure 3

Example of an RNN Used for Residual Evaluation.

4.1 Residual Generation

The MLP used to reproduce the manipulator dynamic behavior has one hidden layer with 49 neurons. The input layer has nine neurons [three joint positions, three joint velocities, and three joint torques measured at (t+Δt)] and the output layer has six neurons [three joint positions and three joint velocities at (t+Δt)]. The training set is formed from 2500 patterns obtained by simulating >90 different trajectories with 25 samples each (at a sample rate of 0.073 s).

Two test sets are used to validate the residual generation. The first set has 5000 patterns obtained in the simulation of 200 random trajectories with a sum of the squared error of the MLP, E=0.019. The second set has 5000 patterns with measurement noise (with variance=0.01) added to the positions and velocities.

4.2 Residual Evaluation

The linguistic variables describing the fuzzified residuals are defined by the following MFs: N, negative residual with a trapezoidal MF; Z, zero residual with a triangular MF; and P, positive residual with a trapezoidal MF.

After many tests on the residuals for different fault sensor situations to achieve a good trade-off between missed detections and false alarms, the following MFs for each residual were selected:

• Residual 1 N1 = [–1 –1 –2e–4 –2e–4] Z1 = [–1.9e–4 0 5e–3] P1 = [6e–3 5.991e–3 1 0.91]

• Residual 2: N2 = [–1 –1 –2.031e–4 –2e–4] Z2 = [–2e–4 0 1e–3] P2 = [0.99e–3 1.001e–3 1 1]

• Residual 3: N3 = [–2 –2 –5e–5 –5e–5] Z3 = [–5e–5 0 0.025] P3 = [0.029 0.022 1 1]

The RNN used in this simulation study is shown in Figure 3. Its training is based on the rules summarized in Table 1, which have been obtained after many simulation tests. The learning operation realized by the back-propagation algorithm converged after 4800 epochs with a sum of the squared error, E=0.0125.

Each row of the inference table represents a rule. For example, rule 2 is expressed as follows:

IF {residual 1 is positive and residual 2 is negative and residual 3 is zero} THEN sensor 1 is faulty.

4.3 Fault Diagnosis

Various simulation tests have been performed to validate the efficiency of this diagnosis scheme, and the results are quite conclusive. For illustrative purposes, only a few fault scenarios are discussed.

4.3.1 Case 1

A non-additive fault is injected on the joint (magnitude fault=0.03). The corresponding residuals are shown in Figure 4. Although a single fault may induce changes in several residuals (here, a fault on joint 1 affects the first residual positively and the second residual negatively at time t = 0.5 s), the decision functions ensure successful detection and isolation of the fault on sensor 1, as shown in Figure 4. The neuro-fuzzy classifier has been trained to recognize the faulty situations from the fuzzified residual patterns according to the rule base given in Table 1.

Figure 4

Residuals with a Non-additive Fault on Joint 1 at t = 0.5 s.

4.3.2 Case 2

A non-additive fault is injected on joint 2 at t = 0.5 s (magnitude fault = 0.02). The residuals and the corresponding decision functions are shown in Figure 5.

Figure 5

Residuals with a Non-additive Fault on Joint 2 at t = 0.5 s.

4.3.3 Case 3

A non-additive fault is injected on joint 1 at t = 0.2 s (magnitude fault = 0.01) and on joint 2 at t = 1.2 s (magnitude fault = 0.02). The residuals and the corresponding decision functions are shown in Figure 6.

Figure 6

Residuals with a Non-additive Fault on Joint 1 at t = 0.2 s and Joint 2 at t = 1.2 s.

4.3.4 Case 4

A non-additive fault is injected on joint 2 at t = 0.2 s (magnitude fault = 0.02) and on joint 2 at t = 1.2 s (magnitude fault = 0.05). The residuals and the corresponding decision functions are shown in Figure 7.

Figure 7

Residuals with a Non-additive Fault on Joint 2 at t = 0.2 s and Joint 3 at t = 1.2 s.

In this section, the sensor faults are detected and isolated successfully using a neuro-fuzzy scheme. In the next section, another approach is used for actuator fault detection and accommodation using a sliding-mode observer.

6.1 Nominal Control

The error is then given by

The dynamics of this error is given by

To determine the nominal control, we propose to refer to the theory of Lyapunov. Consider the following function of Lyapunov defined positive:

V = 12eτe,

then

For a choice of the nominal command given by Eq. (24), V is defined negative.

where Cg(x) is invertible and K is a defined positive matrix.

6.2 Estimation and Compensation for Failure

Once the defect is detected by the module of fault diagnosis, the estimation procedure and compensation is opened. The compensation must, of course, be performed within a minimum time to avoid a significant degradation of system performance. The method of compensation under study falls within the framework of the active approach [1, 21, 23]; it is an additive nominal control to counter the effect of defects. Therefore, we consider the model of the previous system failure, defined by the system of Eq. (35), and we define the function of Lyapunov [Eq. (39)]. V˙ will be given by

The objectives are achieved for an order given by

Thus, the overall command begins in the form

u = U = Un + Ue,

where

and

where U_n is the nominal command and U_c represents the term addendum to compensate for the effect of the actuator or missing component. Nevertheless, the introduction of this term after the appearance of the default requires an estimate of the latter. This can be done through the observer defects, defined in Section 6 by the system of equations

Hence, the additive command will be given by

When the states are not available, we propose to estimate them with a robust observer. It is possible to choose a sliding-mode observer. The estimator of defects will be given by

and the control additive in the form

where x identifies the estimated state.

7.1 Model of the Robot Manipulator

The dynamic model of the robotic system can be derived via either the Lagrangian or Newton–Euler method [2, 9, 22]. Generally, a healthy manipulator arm is described by the following system of differential equations:

where θ,θ˙,θ¨ ∈ Rn denote the vectors of joint positions, velocities, and accelerations, respectively; τ ∈ Rⁿ is the vector of input torques; G(θ) ∈ Rⁿ is the vector of gravitational torque; V(θ,θ˙) ∈ Rn is the vector representing Coriolis and centripetal forces; M(θ) ∈ R^n×n is the inertia matrix whose inverse exists; and μ(θ,θ˙,τ,t) ∈ Rn denotes the unmodeled dynamics. It is assumed that the unmodeled dynamics are bounded.

7.2 Reconfigurable Control Using the Sliding-Mode Observer

Consider the model of the robot described above, given by the dynamic equation

with {θ = [θ1θ2θ3]Tτ = [τ1τ2τ3]TandF(θ,τ).

Include actuators and components defects.

By asking

X1 = [θ1θ2θ3]TandX2 = [θ˙1θ˙2θ˙3]T,

then Eq. (49) becomes

It is possible to rewrite Eq. (50) in the form

We will apply the proposed method in Section 6 for accommodation defects. The system of command tolerant defects is composed of two blocks: one block for the detection and isolation of defects, and another block for the estimation and compensation for the latter. The procedure is as described in this paragraph. It passes by two stages: the first is consecrated to calculating the nominal control law and the second to estimating and compensating for the defects.

The estimation of defects is done using the observer defects defined by

where N is defined a positive matrix; it was chosen to ensure a very fast convergence of the assessed value of default to its true value.

The compensation procedure is aimed to add the nominal command and additive terms given by Eq. (45), upon the occurrence of a default.

Simulations were performed in Matlab to validate our approach. By neglecting uncertainties, we introduced defects actuators at t = 3.5 s. Thereafter, we studied the fault tolerant control in the presence of uncertainties in modeling. It should be noted that similar results were obtained by the introduction of faulty components.

7.3 Simulation Results

For simplicity, a three-link SCARA robot is used in this study. The simulation was conducted using Matlab and Simulink. In the joints (components), the most present type of fault is friction [5]:

Coulomb/stiction:f(θ˙)=αsign(θ˙)Viscous:f(θ˙)=αθ˙

In SCARA manipulators, actuators are generally electric motors and faults may be classified as electric faults:

f(τ) = ατ,−1 < α ≤ K ≤ ∞,

where K is the maximum value of α. The results of simulations in the case of a healthy, but subject to uncertainties, modeling show that residues vary and are moving away from zero. They are therefore sensitive to uncertainties in modeling. This creates false alarms and false detections.

We will now introduce the threshold detection to determine how it is possible to improve the detection of defects and reduce false alarms. The simulation results concerning the third actuator of the SCARA robot are illustrated in Figures 9–13.

Figure 9

Residual 3 with Defect at t = 3.5 s.

Figure 10

Position 3 with Defect at t = 3.5 s.

Figure 11

Velocity 3 with Defect at t = 3.5 s.

Figure 12

Position Error 3 with Defect at t = 3.5 s.

Figure 13

Velocity Error 3 with Defect at t = 3.5 s.