Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator

— The control of any robotic system always faces many great challenges in theory and practice. Because between theory and reality, there is always a huge difference in the uncertainty components in the system. That leads to the accuracy and stability of the system not being guaranteed with the set requirements. This paper presents a novel adaptive single-input recurrent wavelet differentiable cerebellar model articulation controller (S-RWCMAC)-based supervisory control system for an m-link robot manipulator to achieve precision trajectory tracking. This adaptive S-RWCMAC-based supervisory control system consists of a main adaptive S-RWCMAC, a supervisory controller, and an adaptive robust controller. The S-RWCMAC incorporates the advantages of the wavelet decomposition property with a CMAC fast learning ability, dynamic response, and input space dimension of RWCMAC can be simplified; and it is used to control the plant. The supervisory controller is appended to the adaptive S-RWCMAC to force the system states within a predefined constraint set and the adaptive robust controller is developed to dispel the effect of the approximate error. In this scheme, if the adaptive S-RWCMAC can not maintain the system states within the constraint set. Then, the supervisory controller will work to pull the states back to the constraint set and otherwise is idle. The online tuning laws of S-RWCMAC and the robust controller parameters are derived from the gradient-descent learning method and Lyapunov function so that the stability of the system can be guaranteed. The simulation and experimental results of the novel three-link De-icing robot manipulator are provided to verify the effectiveness of the proposed control methodology. The results indicate that the proposed model has superior accuracy compared to that of the Standalone CMAC Controller. The parameters of the average squared error in the S-RWCMAC - based 3 robot joints are lower than those of the Standalone CMAC Controller by 0.023%, 0.029%, and 0.032%, respectively.


INTRODUCTION
Fuzzy Logic Control (FLCs) are deployed in complex, high-order systems that require the use of all states.These state variables are used to represent the contents of the rule premises.Therefore, creating a complex network of rules becomes a significant challenge in designing and implementing FLCs.This creates distinct difficulty when requiring many control rules and requires considerable effort from experts.However, in complex higher-order systems, we often must increase the number of state variables.That means increasing the number of control rules.This not only takes time and effort in the process of creating rules but also requires a deep understanding of the system and control rules on the part of experts.Therefore, deploying FLCs in complex, highorder systems become a difficult task.To overcome this challenge, a careful assessment of the effectiveness and complexity of creating and implementing control rules should be conducted.Research and development of advanced methods to reduce the number of rules and effort required to implement FLCs is an important step to overcome the limitations of this approach.To address these issues, singleinput Fuzzy Logic controllers (S-FLC) were proposed for the identification and control of complex dynamical systems [1]- [5].As a result, the number of fuzzy rules is greatly reduced compared to the case of conventional FLCs, but its control performance is almost the same as conventional FLCs.
Cerebellar model articulation controller (CMAC) was proposed by Albus in 1975 [6] for the identification and control of complex dynamical systems, due to its advantage of fast learning property, good generalization capability, and ease of implementation by hardware [7]- [10].The conventional CMACs are regarded as non-fully connected perceptron-like associative memory networks with overlapping receptive fields which used constant binary or triangular functions.The disadvantage is that their derivative information is not preserved.For acquiring the derivative information of input and output variables, Chiang and Lin [11] developed a CMAC network with a differentiable Gaussian receptive-field basis function and provided the convergence analysis for this network.The advantages of using CMAC over neural networks in many applications were well documented [12]- [15].
Most control systems are developed in the literature based on the CMAC which is used to approach the nonlinear mapping [16]- [20], in which the input dimension of CMAC includes all state variables of the system.As a result, many adaptive approaches are also rejected as being overly computationally intensive because of the real-time parameter identification and required control design.To deal with these problems, other control schemes are also proposed in [21]- [25] by combination with conventional control techniques (sliding model control, etc.) so that the dimension of the input space is reduced.Recently, there are some researchers [26]- [30] who have reported very good results based on the CMAC Journal of Robotics and Control (JRC) ISSN: 2715-5072 439 Thanh Quyen Ngo, Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator control system.In [31]- [34], the proposed single-input CMAC controllers are not only solely used to control the plant, so the input space dimension can be simplified, and no conventional controller.However, the disadvantage of the proposed CMAC control system adopts two learning stages, an off-line learning stage, and an on-line learning stage.In [35], the later proposed controller is developed the same in [32], but this control scheme not only overcomes disadvantages in [33] but also the stability of the control system can be guaranteed.However, the major drawback of the above single-input CMACs is they belong to static networks.
Many studies have been conducted on the application of neural networks (NNs) to predict, recognize, and control dynamic systems [36]- [40].One of the most outstanding advantages of NN is its ability to approximate arbitrary linear or nonlinear systems through learning.Based on that advantage, NNs are used to approximate mathematical models of control systems.Structurally, NNs can be classified primarily into transmission neural networks (FNNs) [41], [42] and regression neural networks (RNNs) [43], [44].RNNs have capabilities superior to FNNs, such as status response and information storage capabilities [43], [44].Because the RNN has a feedback loop, it captures the state of the system through error, so the RNN represents better control performance.However, regardless of FNN or RNN, the learning process is slow because all weights are updated in each study cycle.Therefore, the effectiveness of NN is limited in math problems that need to be learned online.The cerebellar model articulation controller (CMAC) is widely applied to control the closed loop of complex dynamic systems due to its fast-learning characteristics, good generalization capabilities, and simple computations compared to multi-user perceptrons with back-propagation algorithms [45], [46].The CMAC is a perceptron-like associative memory network that is not fully bound to overlapping receiving fields.The application of CMAC is not limited to control problems but also to the modeler's function approximation.The CMAC network has proven that it can approximate a nonlinear function with any desire.The advantages of using CMAC over conventional NN in many practical applications have been presented in recent literature [47]- [50].However, the main disadvantage of current CMACs is that their application domain is limited to static.Over the past decade, much research has been conducted on the applications of wavelet neural networks, combining the ability to learn from the processes of artificial neural networks and wavelet separation capabilities [51]- [53] to identify and control dynamic systems [54]- [59].In [54], wavelet networks were proposed as an alternative to FNN for approximate arbitrary nonlinear functions based on wavelet transform theory, and the backstream algorithm was adapted to train wavelet networks.Many successful applications have been implemented based on neural wavelet networks (WNNs) that combine network learning and wavelet dissociation [55]- [58].Unlike conventional NN networks, WNN member functions are spatially localized wavelet functions.Therefore, WNN networks are capable of learning more efficiently than conventional NN networks for system control and identification as demonstrated in [55]- [57].Therefore, WNN networks have been of considerable interest in applications to handle control systems with uncertain and non-linear mathematical models as presented in [57], [58].Zhang et al. in [59] described wavelet-based neural networks for learning and estimating mathematical models.The structure of this network is like that of the radial base function network except that the radial functions are replaced by orthogonal scaling functions.From a function representation point of view, traditional radial base function networks can be represented under any function in extended space.However, it is redundant.That means that a given function's radial base function lattice representation is not unique and probably not the most efficient.
In this paper, by combining the fast learning property of CMAC, the capability of the wavelet decomposition property, by including a delayed self-recurrent unit in the association memory space, and based on [1], [2], the system tracking error  ∈   is transformed into a single variable, termed the signed distance   ∈   , we propose a novel adaptive singleinput recurrent wavelet CMAC (S-RWCMAC)-based supervisory control system which presents a dynamic S-RWCMAC with single-input for three-link De-icing robot manipulator to achieve the precision trajectory tracking.This control system consists of an adaptive S-RWCMAC, a supervisory controller, and an adaptive robust controller.The S-RWCMAC is the main controller which is used to mimic the ideal controller through learning and the adaptive robust controller is developed to dispel the effect of the approximation error.The online tuning laws of S-RWCMAC parameters are derived in the gradient-descent learning method which can be caused by the instability-controlled system, especially in the transient period.So, the supervisory controller is appended to the adaptive S-RWCMAC to force the system states within a predefined constraint set, if the adaptive S-RWCMAC can not maintain the system states within the constraint set.Then, the supervisory controller will work to pull the states back to the constraint set, otherwise is not.
• The use of S-RWCMAC in de-icing robot systems on power lines has several important contributions as follows: High performance: The S-RWCMAC is specifically designed to work on power lines and ice breakers.It can reach and work in difficult and dangerous positions that humans cannot perform.The use of this robot increases work efficiency and reduces the time required to restore the electrical system.
• In addition, the supervisory and the adaptive robust controllers are proposed to append to the S-RWCMAC in order to get rid of the approximation error and maintain the system stability.
• Safety for humans: The use of S-RWCMAC minimizes the hazards and risks associated with humans having to work on high voltage poles and in extreme weather conditions.This robot can perform icebreaking tasks automatically and accurately, thereby reducing the risk of work accidents.
• Flexibility and remote controllability: The S-RWCMAC can be controlled remotely, allowing remote management and operation of icebreaker ISSN: 2715-5072 440 Thanh Quyen Ngo, Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator operations on power lines.This minimizes human intervention and provides flexibility in performing various tasks.
In summary, the use of S-RWCMAC in ice breaker robots on power lines brings many important contributions such as increasing work efficiency, ensuring human safety, saving costs and providing flexibility during control.This paper is organized as follows: System description is described in Section II.Section III presents the S-RWCMACbased supervisory control system.Numerical simulation and experimental results of a three-link De-icing robot manipulator under the possible occurrence of uncertainties are provided to demonstrate the tracking control performance of the proposed S-RWCMAC system in Section IV.Finally, conclusions are drawn in Section V.

II. SYSTEM DESCRIPTION
In general, the dynamic of an m-link robot manipulator may be expressed in the Lagrange in the equation (1).
( ) Where (), () are nonlinear dynamic functions which are difficult to determine exactly or can not even obtain.So, we can not establish a model-based control system.In order to with this problem, here we assume that actual value () and () can be separated as nominal part denoted by  0 (),  0 (), in which  0 () is assumed to be positive, differentiable, and  0 −1 () exists for all .(, ) is represented as the unknown lumped uncertainly and  = [  , ̇]  is a vector that represents the joint position and velocity.Finally, the system (2) can be rewritten as (3).
The control problem is to force () ∈   , to track a given bounded reference input signal   () ∈   .Let () ∈   be the tracking error as (4).
and the system-tracking error vector is defined as (5).
Where n is the order of the nonlinear system.If the nominal parts  0 (),  0 () and the uncertainly (, ) are exactly known, then an ideal controller can be designed as (6).By substituting the ideal controller ( 6) into (3), the error dynamic equation is given as (7).
It is obvious that errors will asymptotically tend to zero if the gain matrices of  = [ 1 , ⋯ ,   ]  ∈   is determined so that the roots of the characteristic polynomial () =  ̈+  1  ̇+  2  +  3 lie strictly in the open left haft of the complex plane.However, the ideal controller in (6) can not determine, because of (, ) is exactly unknown for practical applications.So, to this problem, a proposed adaptive S- RWCMAC-based supervisory control system is shown in Fig. 2 which is described in the following sections.

III. ADAPTIVE S-RWCMAC-BASED SUPERVISORY CONTROL SYSTEM
The architecture of the adaptive S-RWCMAC-based supervisory control system is shown in Fig. 2 which consists of adaptive S-RWCMAC.The supervisory controller, and the adaptive robust controller with the equation (8).
Where  − is the main controller based on the S-RWCMAC which is used to mimic the ideal controller in (6) and   is the supervisory controller, which can be used to stabilize the states of the controlled system within a predefined constrain set and the adaptive robust controller   is utilized to compensate for the approximation error between the ideal controller and  − .

A. Brief of the S-RWCMAC
An S-RWCMAC is proposed based on [19] and is depicted in Fig. 3.This S-RWCMAC is composed of an input space, an association memory space, a weight memory space, an output.The signal propagation and the basic function in each space are introduced as follows.
1. Input Space   : assume that each input state variable   can be quantized into   discrete states and that the information of a quantized state is distributive stored in   (  ≥ 2) memory elements.Therefore, there exist   + 1 individual points on the   axis.Fig. 4 depicts the schematic diagram of one dimensional S-RWCMAC operations with   = 3 and   = 7 for  = 1, 2 ⋯  where the equidistant quantization scheme is used to partition the input space [−1, +1].This simple example shows that the input space is quantized into three discrete regions, called blocks.For instance, there are three blocks, namely, A, B, and C, in the first layer.By shifting each block to a small interval, different blocks can be obtained.For example, D, E, and F in the second layer are possible shifted regions.With this kind of decomposition, one can imagine that there are   = 3 layers of blocks.Each state is covered by   = 3 different blocks.
The S-RWCMAC associates each block to a physical memory element.Information for a quantized state is distributed and stored in memory elements associated with blocks that cover this state.Note that a state will share some memory elements with its neighboring states, but any two states will not correspond to the same set of blocks.Moreover, the entire memory size that is equal to the number of blocks and denoted by  ℎ is determined by  ℎ =   +   − 1. See the block division shown in Fig. 4 for instance; the total number of memory elements is 9.In this space, each block performs a receptive-field basis function, which can be defined mother wavelet.The first derivative of the basic Gaussian function for each block is given here as a mother wavelet which can be represented as (9).,   represents the reception-field basic function for th block of the th input,   is a translation parameter, and   is dilation.In addition, the input of this block can be represented as (10).
Where   is the recurrent gain,  denotes the time step and   ( − 1) denotes the value of   () through a time delay.The input of this block contains the memory term  ( − 1), which stores the past information of the network and presents the dynamic mapping.

Output Space O:
The output of S-RWCMAC is the algebraic sum of the firing element with the weight memory, and is expressed as (11).
Where   denotes the weight of the jth block,   =   (  ),  = 1, 2, ⋯  ℎ is the index indicating whether the th memory element is addressed by the state involving   .
Since each state addressed exactly   memory elements, only those addressed   , s are one, and the others are zero.

B. On-line Learning Algorithm
The central part of the learning algorithm for an S-RWCMAC is how to choose the weight memory   ,   is a translation parameter,   is dilation of the wavelet functions and   is a recurrent gain.For achieving effective learning, an on-line learning algorithm, which is derived using the supervised gradient descent method, is introduced so that it can in real-time adjust the parameters of S-RWCMAC.
According to [1], [2], the system tracking error (  , ̇) ∈   is transformed into a single variable, termed the signed distance   ∈   , which is the distance from an actual state (  , ̇) ∈   to the switching line as shown in Fig. 5 for a 2-D input.The switching line is defined as (12).
Where  −1 is a constant.Then, the signed distance between the switching line and the operating point (  , ̇) ∈   can be expressed by the equation (13).
Define a cost function (  ()) = With this representation of the S-RWCMAC system, it becomes straightforward to apply the backpropagation idea to adjust the parameters.The weight memory   and the translations   and dilations   of the mother wavelet function are updated by the following: 1.The updating law for the th j weight memory can be derived according to (16).
Where   is the positive learning rate for the output weight memory   , the connective weight can be updated according to the equation (17).
2. The translations and dilations of the th j mother wavelet function can be also updated according to (18).
Where   ,   are positive learning rates for the translation and the dilation parameters.The translation and dilation can be updated as (20) and (21).
Where   is the learning rate, the recurrent gain can be updated by the equation (23).

C. Supervisory Controller
The design of a supervisory controller is necessary in case of divergence of states.This controller is used to pull the states back to a predefined constraint set.The supervisory controller fires only when the system states leave the predefined constraint set.When the system states stay within the constraint set, only the adaptive S-RWCMAC will be utilized to approximate the ideal control law.The supervisory controller is presented as follows.From (3) and (6) and using (8), an error equation is obtained as (24). Where, The Lyapunov function is defined as (25).
Where  ∈  × is a symmetric positive definite matrix which satisfies the Lyapunov equation (26).
And  ∈  × is a positive definite matrix.Take the derivative of the Lyapunov function and use ( 24) and ( 26), then obtained equation (27).
Based on Assumption 1 and observing (27), the supervisory controller is designed as (28).
( ) Where sgn( ) is a sign function, and the operator index as in (29).
In which  ̄ is a preset positive constant.Substituting ( 6) and ( 28) into (27) and considering the case  = 1, it is obtained that equation (30).
Using the supervisory controller   shown in (28), the inequality  ̇ ≤ 0 can be obtained for nonzero value of the tracking error vector  when   >  ̄.As the results from (30), the supervisory controller is capable to drive the tracking error to zero.

A. Adaptive Robust Controller
The S-RWCMAC approximation ( 11) is used to approximate the ideal controller (6) through learning.By the universal theorem, there exists an S-RWCMAC  − to approximate  * (31) [60].
* =  − +  (31) Where  denotes the approximation error which is assumed to be bounded by 0 ≤ ‖‖ ≤ , if  is assumed to be a positive constant during the observer.Then, the conventional robust controller is designed as (32).
However, the parameter variations of the controlled system are difficult to measure, and the exact value of the load disturbance is also difficult to know in advance for practical applications.Therefore, the error-bound estimation needs a continuous prediction by the proposed fuzzy logic controller (FLC) bound observer.In the general FLC, as shown in Fig. 6. every fuzzy rule is composed of an antecedent and a consequent part, a general form of the fuzzy rules can be represented as (33).
Where   is the input dimension,   and   are the rule for th input variable,   is the output weight in the consequent part and   =     is the number of the fuzzy rules.Here, the membership functions for the input are chosen to be a triangular type as shown in Fig. 6.
Where  = [  1  2 ⋯    ]  ∈    is a firing strength vector of rule and  = [  1  2 ⋯    ]  ∈    is the consequent parameter vector which is adjusted by the adaptive rule.By the universal theorem [60], there exists an optimal FLC in the form of (34) such that.
Where  * is the optimal weighting vector that achieves the minimum approximation error and  is the approximation error of the FLC and assumed to be bounded by || ≤ .
Replacing  by  ̂ in (32), the robust controller can be represented as (36).
To compensate for the approximation error of the FLC   is developed as (37).
Theorem 1: Consider an n-link robot manipulator expressed (2).If the adaptive S-RWCMAC-based supervisory control law is designed as (8) in which the supervisory control law is designed in (28), the S-RWCMAC is presented in (11) with the adaptive laws of the S-RWCMAC are designed as ( 16), ( 18), ( 19) and ( 22), and the robust controller is developed as (37) with the estimation law in ( 34), ( 39) to (40), then the stability of the proposed S-RWCMAC-based supervisory control system can be ensured.
Where  ̃=  * −  ̂,  ̃=  −  ̂,   and   are approximation error of fuzzy compensation, estimation error, and learning rates for the fuzzy compensator and the error estimator, respectively.By differentiating (42) with respect to time and using ( 26) and ( 38), we can obtain equation (43).
In addition, ℎ ̇() is bounded; thus, Barbalat's lemma can be shown that  →∞ ℎ() = 0.It can imply that  will be converging to zero as time tends to be infinite.As a result, the stability of the proposed adaptive S-RWCMAC-based supervisory control system can be guaranteed.

A. Simulation Results
A three-link De-icing robot manipulator as shown in Fig. 1 is utilized in this paper to verify the effectiveness of the proposed control scheme.The detailed system parameters of this robot manipulator are given as link mass  1 ,  2 ,  3 (), lengths  1 ,  2 (), angular positions  1 ,  2 () and displacement position 3 ().
The parameters for the equation of motion (1) can be represented as (47).]
(a) ( The simulation results indicate that the high-accuracy trajectory tracking responses can be achieved by using the proposed adaptive S-RWCMAC-based supervisory control system for periodic step reference trajectory.However, the performance measure comparison of the proposed S-RWCMAC control system with the standalone CMAC control system which has been proposed in [19] is also shown in Table I.This table shows that, for the adaptive S-RWCMAC control system, the root mean square errors are 0.038%, 0.023%, and 0.012% for periodic step commands for each link, respectively.Moreover, comparing the proposed S-RWCMAC control system with the standalone CMAC control system, the root means square errors have been reduced by about 0.023%, 0.029%, and 0.032% for periodic step commands for each link, respectively.This indeed confirms the performance improvement of the proposed S-RWCMAC control system.
Based on the given data, we can analyze some important points as follows: 1. Control performance evaluation: RMS is a measurement commonly used to evaluate the accuracy and efficiency of a control system.The lower the RMS value, the smaller the error and the better the control performance.

Comparison between Standalone CMAC Controller and S-RWCMAC Controller:
Based on the given data, we see that the S-RWCMAC Controller has a lower RMS value than the Standalone CMAC Controller for all links.This shows that the S-RWCMAC Controller has better control in reducing errors and achieving the desired output position.At the same time, it shows that the S-RWCMAC Controller can achieve a more accurate response and reducing errors in controlling links.However, for a more comprehensive and accurate assessment of the performance of these controllers, additional factors such as stability, response speed, and ability to respond to interference and fluctuations in the control system should be considered.

B. Experimental Results
The hardware block diagram of the control system is implemented to verify the effectiveness of the proposed methodologies and is shown in Fig. 8(a).An image of a practical experimental control system for De-icing robot consists of three manipulators and is shown in Each joint of the manipulator is derived by the "EC─**" type MAXON DC servo motors, which are designed by Switzerland Company, and each this motor contains an encoder.Digital filter and frequency multiplied by circuits are built into the encoder interface circuit to increase the precision of position feedback.The DCS303 is a digital DC servo driver developed with DSP to control the DC servo motor.The DCS303 is a micro-size brush DC servo drive.It is an ideal choice for this operating environment.Two DC servo motor motion control cards are installed in the industrial personal computer, in which, a 6-axis DC servo motion control card is used to control the joint motors and a 4-axis motion control card is used to control the drive motors.Each card includes multi-channels of digital/analog and encoder interface circuits.The name of the model is DMC2610 with a PCI interface connected to the IPC.The DMC2610 implements the proposed program and executes it in real time.Considering that the control sampling rate Ts = 1 ms is too demanding for the hardware implementation, Ts = 10 ms is thus considered here.
The experimental parameters of the proposed S-RWCMAC-based supervisory control system are selected in the same simulation.In this section, the control objective is to control each joint angle of the three-link De-icing robot manipulator to move different sinusoidal commands [ 1 ,  2 ,  3 ] = [2 ( 0.2), ( 0.2), ( 0.2)] and the initial conditions of system are given as According to these experimental results in Fig. 9 of proposed S-RWCMAC-based supervisory control system due to sinusoidal reference trajectories; it is shown that highaccuracy tracking performance of proposed S-RWCMAC-based supervisory control system can also be archived for sinusoidal reference commands.For the experimental results, the performance measure comparison of the proposed S-RWCMAC control system with the standalone CMAC control system which has been proposed in [19] is also shown in Table II.This table shows that, for the adaptive S-RWCMAC control system, the root mean square errors are 0.078%, 0.055%, and 0.043% for sinusoidal commands for each link, respectively.
(i) Fig. 9. Experimental position responses, tracking errors of the proposed adaptive S-RWCMAC-based supervisory control system at joints 1, 2 and 3 Moreover, comparing the proposed S-RWCMAC control system with the standalone CMAC control system, the root means square errors have been reduced by about 0.023%, 0.027%, and 0.011% for sinusoidal commands for each link, respectively.This indeed confirms the performance improvement of the proposed S-RWCMAC control system.

CONCLUSIONS AND DISCUSSION
This study has successfully implemented an adaptive S-RWCMAC-based supervisory control system for the threelink De-icing robot manipulator to achieve high-precision position tracking performance, which plays a very crucial role in the field of robotics.By using the S-RWCMAC structure, the system is capable of learning and adapting quickly to the environment.At the same time, the system is also capable of adjusting the parameters of the model to achieve optimal results.One of the outstanding advantages of the system is the combination of wavelet neural networks and the fast-learning capabilities of CMAC.The use of decay properties?(locality) and fast learning capabilities helps reduce data size and increase the processing speed of S-RWCMAC, creating a more flexible and effective control method.
The adoption of the adaptive controller and the durable controller is also shown to ensure safety, avoidunwanted states, and minimize the impact of errors on system performance.In addition, the online tuning laws of S-RWCMAC parameters and error estimation of the adaptive robust controller which are derived from the gradient-descent learning method and the Lyapunov function guarantee the stability of the system.The simulation and experimental results indicate that the proposed S-RWCMAC-based supervisory control system can achieve favorable tracking performance for different reference commands.
With its good tracking performance, this monitoring control system is promising to a variety of automated systems such as mobile robots, AC servo systems, etc.
For a more comprehensive evaluation of the proposed adaptive S-RWCMAC-based monitoring control system, more research of the detailed structure of S-RWCMAC can be conducted.
The S-RWCMAC controller is a high-level algorithm in the field of control for dealing with uncertain components in nonlinear systems.These algorithms can learn from system errors and adjust parameters to optimize control performance.Thereby bringing adaptability and flexibility to the system.By dealing with uncertain components, the S-RWCMAC has the potential to improve the control system's performance.However, like other algorithms, they also face some limitations and challenges.
First, calculating and processing data in the control system requires a lot of resources to calculate.Ensuring high performance and fast system response time requires good and efficient data processing.This can pose challenges in terms of complex algorithms.
Second, environmental conditions may create other limitations and challenges to the performance of the control system.Interference from outside and inside the system can significantly affect the stability and operation of the system.Building control systems to deal with these factors is no small challenge.
Despite the limitations and challenges, the advancement and application of the S-RWCMAC control system remain very promising.The extension and application of S-RWCMAC can be adapted to handle complex systems and meet control system requirements.
However, the research and development of new algorithms are necessary to improve the performance and application of control systems.Continued investment in control technology research and development is critical to making progress in this area.Especially exploiting the potential of the S-RWCMAC control system.In summary, CMAC and S-RWCMAC are important and promising technologies in the field of control.However, to overcome limitations and challenges, it is necessary to invest in further research and development and to create optimal hardware and software architecture solutions for the effective implementation of CMAC and S-RWCMAC control systems in practice.

Fig. 1 .
Fig. 1.Architecture of three-link De-icing robot manipulator , Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator

Fig. 6 .
Fig. 6.Fuzzy rule membership functionsIn this paper, the defuzzification of the output is obtained by the height method as(34).

) Fig. 7 .
Simulation position responses and tracking error of the proposed adaptive S-RWCMAC-based supervisory control system at links 1, 2 and 3 , Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator Fig 8(b).The left and right manipulators have three-link with two revolute joints and a prismatic joint.End-effectors of each manipulator have attached the motion structure to move the De-icing robot on the power line and the de-icing device.During normal operating conditions, the left and right manipulators are only operating.The manipulator has only two joints a revolute joint and a prismatic joint.It only works when the De-icing robot avoids obstacles on the power line.In general, the operation of De-icing robot is very complex.In this paper, we consider only the three-link De-icing robot manipulator for proposed methodologies while the other manipulator is the same.

Fig. 8 .
Fig. 8. IPC-based De-icing robot position control system a) Block diagram of three-link De-icing robot manipulator control system, b) image of practical control system, c) image of special robot laboratory of power industry Fig. 8b

Fig. 8c
Fig. 8c , Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator , Adaptive Single-Input Recurrent WCMAC-Based Supervisory Control for De-icing Robot Manipulator V.

TABLE I .
PERFORMANCE MEASURE OF S-RWCMAC AND THE STANDALONE CMAC APPROXIMATION

TABLE II .
PERFORMANCE MEASURE OF S-RWCMAC AND THE STANDALONE CMAC APPROXIMATION