A Comparative Study Between Convolution and Optimal Backstepping Controller for Single Arm Pneumatic Artificial Muscles

— This study was based on the dynamic modeling and parameter characterization of the one-link robot arm driven by pneumatic artificial muscles. This work discusses an date- to-up control design based on the notion of a conventional and optimal backstepping controller for regulating a one-link robot arm with conflicting biceps and triceps positions supplied by pneumatic artificial muscles. The main problems found in systems that utilize pneumatic artificial muscle as actuators are primarily the large uncertainties, non-linearities, and time-varying features that severely impede movement performance in tracking control. In consideration of the uncertainty, high nonlinearity, and external disturbances that can exist during the motion. Lyapunov-based backstepping control technique was utilized to assure the stability of the system with improved dynamic performance. The bat algorithm optimization method is utilized in order to modify the variables used in the design of the controller to enhance the efficiency of the suggested controller. According to the conclusions, a quantitative comparison of the response in the PAM actuated the arm model in the current study and earlier investigations with the Backstepping controlled system revealed fair agreement with a variation of 37.5% from the optimal classical synergetic controller. In addition, computer simulations were utilized in order to compare the effectiveness of the proposed conventional controls and the optimal background. It has been proven that an optimal controller can control the uncertainties and maintain the controlled system’s stability


I. INTRODUCTION
Pneumatic artificial muscles (PAMs) appear to hold a lot of promise in industrial applications for innovative robot and manipulator models [1]. PAMs are used in contemporary robotic systems because they typically provide high-speed action skills, an uncomplicated working mechanism, and safety operations. The power produced by PAM actuators doesn't rely just upon pressure yet in addition on the condition of expansion, which adds one more wellspring of the spring-like way of behaving [2]. Since multilayer structures are the central component of these actuators, these PAMs that mimic human muscle movement are lightweight. [3], [4] Biomechanics, bio-advanced mechanics, mechanical technology, and counterfeit appendage substitution have all utilized PAM actuators. Further, the PAMs are noiseless gadgets, they can be utilized in emergency clinic medicines to clamor delicate patients [5]. Due to their powerful/volume proportions, PAMs don't need a stuff framework to help power when contrasted with engine actuators. PAMs are exceptional-lightweight natural responsiveness, excellent specific work, elasticity [6], and are valuable for the regular recurrence of biped motion because of their gracefulness [7], [8] PAMs have numerous disadvantages that characterize them. The PAMs combative structure is one disadvantage that can be addressed as compared to other actuators. Another major issue is the failure to control PAMs due to their high time-varying, nonlinear, and uncertain parameter structure [9], [10] This is a result of the mechanics of the system containing several uncertain, non-linear, and unknown factors that prevent the development of an effective actuator tracking controller [11]. Additionally, they are quite sensitive. The operational ranges of the PAM systems are severely restricted by parameters affecting the systems such as viscosity, temperature, and supply pressure [12], [13].
Due to the importance of the operation of PAMs and as they mimic real muscles, PAM is a useful device for implementing the humanoid [14]. The control of pneumatic muscles is difficult because the physical parameters are nonlinear and time-varying [15].
To address difficulties related to the control of mechanical systems driven by aerobic muscles, several researchers have put forward various control solutions. Where PAM-actuated devices employ the most recent control methodologies.
Lilly [16] recommended sliding-mode adaptive controller for planar pneumatic muscle-powered robot arm. The actual configurations of the nonlinear functions, including the system mechanical parameters, such as link weights, lengths, and inertias, are required by this adaptive controller. In addition, their wide operating range has been significantly restricted due to their extreme sensitivity to parameters affecting the operation of PAM systems, like the temperature, viscosity, and applied pressure.
Scaff et al. [17] proposed a McKibben PAM-actuated with the regular Proportional-Integral-Derivative (PID) controller, position control of a one-degree-of-freedom (1-DOF) system. The Simulated Optimization Algorithm (SOA) is used to modify the PID controller's parameters in the Choi et al. [12] A method for controlling PAM-powered robots has been introduced that would replace the proportional pressure regulator (PPR) with a control unit consisting of several micro encoders and pressure switches to address capacitance problems associated with PPR. new controller may be able to reduce capacity requirements, but at the sacrifice of accuracy, according to the results of the trial evaluation. The controller is only there to save storage and relies on an on-off mechanism while ignoring the unpredictability in the system parameters.
Enzevaee et al. [18] proposed Active Force Control (AFC) system based upon Fuzzy Logic (FL) controllers to track the control of a single-link robotic arm. This outcome from simulation and experimentation was used to evaluate the proposed control method's sustainability and reliability in calming the interruptions that it was subjected to. The primary conclusion that has been presented is that inputoutput gain has been used to mimic the robot arm's dynamics. Furthermore, the tracking controller's use of a PID controller was unable to offset the system parameter uncertainty.
Al-Jodah and Khames [19] described a 1 st order and 2 nd order sliding mode control (SMC) to track the angular displacement of a single-link robotic arm powered by two PAMs. The ability of the presented controllers for reducing chattering in associated control outputs as well as their robustness against uncertainties in the system variables have been investigated. However, mainly concentrated on finding a solution to the chattering problem commonly associated with SMC architecture.
Medrano-Cerda et al. [20] designed a bi-muscular pneumatic muscle activator system, adaptive controller. Adaptive pole-placement control has been used to construct the PAM system's control strategy. The suggested adaptive scheme has been based upon indirect control strategies, where mathematical model and system variables are estimated based on on-line input and output data collected using the suggested model architecture, although the controller can provide viable accuracy and result in a high ratio of power to weight.
Boudoua et al. [21] To control the PAM-operated robotic arms and decrease the chattering in the signal out of the control, an NN-based Twisting Sliding Mode Controller (TSMC) was introduced. This study utilized two methods two-layer Neural networks (NN) and on-line adaptive learning in order to simulate nonlinear and unidentified robot dynamics. The present work was unable to totally eradicate chattering in the output signal from the control, and using NN structures up to approximation may reduce a controller's performance except if the required quantity and kind of activation functions are applied.
Jahanabadi [22] studied the implementation of an integrated regulator for the trajectory tracking of a PAM-actuated 2-planar link manipulator based upon Active Force Control and FL (AFCFL). The FL, is controlled by an outer loop PID controller and is used to choose the optimal structure of the inertial matrix requested for the AFC mechanism for the robot arm. In addition, the principal tracking controller was a PID controller and a fixed gain has been used in order to simulate the dynamic model, which is significantly different from the original model.
Previous research has shown that, despite significant advancements in PAM's development and various control strategies, there is still considerable work to be completed. The earlier research employed FL, NN, control based on optimization, nonlinear control based on SMC, or hybrid nonlinear control, which are all examples of advanced control systems. However, these controllers could not solve all major issue such as the uncertainty, non-linearity, and the chattering that appear in the output signal of the system.
It is worth mentioning that the PAM-actuated manipulator's architecture varies between studies. In contrast to earlier research. the Backstepping Control (BSC) technique is used in this study to develop a controller for tracing and controlling of the PAM--actuated one-link robot arm movement. The BSC theory is based on space -state theories, that are used in the development and management of extremely complicated and interconnected nonlinear systems. The BSC method is based upon a control approach that is suitable for a specific class of non-linear systems.
To deal with all these paramount problems, an adaptive control strategy is suggested for controlling the PAMactuated one-link robot arm, which can transact effectively with the impact generated by parametric uncertainties in actuating muscles The dynamic performance of this controllable system is directly impacted by the BSC's design variables. The Bat Algorithm has been used in order to enhance the systems with uncertainties in actuation muscles of PAM modeling, reduction of the chattering, and modify those parameters because the trial-and-error approach for calculating those components is challenging, time-consuming, and does not produce the optimal dynamic stability response, it is necessary for improving the dynamic performance regulation and detection. Bats served as the inspiration for this echolocation algorithm, which was originally created by Yang [23].
This research aims to develop BSC to maintain and coordinate the tracking of desired motion while reducing chattering, non-linearity, and uncertainty, in the manipulator arm that is operated by PAM in the system to maintain the stability of the system. This paper's contribution is to develop a control strategy based on BSC theory for tracing the motion of a single-link robot arm actuated by pneumatic artificial muscles while taking into account parameter uncertainties in the muscles. The suggested technique is tenacious and capable of preventing chattering while also compensating for the parametric uncertainties.
Using the bat algorithm with the proposed controller design parameters to make progress in implementation can be This paper will contain the following sections in a sequence: • The dynamics and control model contains the derivation of the mathematical model of one link arm actuated by PAMs with suggestion BSC.
• The optimization of the system using the bat algorithm.
• Results and discussion show the simulation results and discussion of the control system and model response.
• Finally, the Conclusion section concludes the paper.
The methodology of this paper is described in the block diagram that is shown in Fig. 1. and it shows the sequence of portraying the contents of this research.  Fig. 1 shows a model of a PAM type and dimensions of the fluidic muscle that this study will concentrate on the fluidic muscle (DMSP-20-100N-RM-CM) from the FESTO Company. Because it responds more quickly than other types and movements like a natural muscle, its work efficiency is up to 50% closer to the biological muscles. Theoretical Fluidic Muscle force at maximum operating pressure is 1500N, the mode of operation is single-acting mode and pulling mode, and the maximum working load freely suspended is 80Kg. The operating pressure of this kind is between 0 MPa and 0.6 MPa [24].   Before starting the design control for the system utilizing PAM, it is required to 1 st develop a mathematical model of the system that accurately reflects real muscle demeanor. The PAM system is able to be examined and its connected controller is designed to suit the implementation prerequisites. represents pneumatic muscle extension (m), and represents the muscle contraction (m). The PMs are attached to elbow at point , which is rotational axis away from the joint. is distance between joint and the load's center of mass.

II. DYNAMICS AND CONTROL MODEL
The extent to which pneumatic muscles extend and muscles contract can be represented respectively by Eq. (1) and (2) [16], [25] The movement of the wrist shown an angle = −1 ( / ) with the triceps corps. Within the same angle , the wrist is authorized to twist. The angle = 0 corresponds to the wrist in a descending position, whereas angle = represents a case that the wrist is positioned extremely in which the triceps muscle's counterclockwise torque is represented as: where (. ) and (. ) represent developed forces from triceps and bicep of the PAMs, respectively, and r represents the radius of the pulley. The produced (. ) and (. ) may be described by the following dynamic PAM model [16], [25]: Where bicep coefficients of viscous friction is ( ), the represents distance of the arms between the mass's centroid and joint, ( ) represents triceps coefficients of viscous friction, ( ) denotes bicep spring coefficients (N/m), ( ) denotes triceps spring coefficients (N/m). ( ) represents the force that is exerted by PAM in bicep case, ( ) denotes force that has been exerted by PAM in triceps situations, a represents distance between joint axis of rotation and PAMs attached point (A), ( ), ( ) and ( ) represent bicep PAM force, spring and viscosity coefficients, respectively, and those can be expressed in the following form: As well, ( ), ( ), and ( ) characterizes triceps of the PAM force, spring, and viscosity coefficients, so that the associated formulas explain them: It is important to point out that coefficient relying on whether a muscle is already in compressed mode or stretched mode, which is, one have varied coefficients of the triceps and bicep ( ) and ( ). Therefore, by combining the torques described by Eq. (3) and Eq. (4), one can find the dynamics motion equation: where = 2 describes the moment of mass inertia about the elbow and latest term ( * * * ) has been adjusted to take into consideration the mass gravity's counterclockwise torque on forearm. So can achieve the following by substituting Eq.
It is provided time derivatives of PM extension and contraction may be described, respectively, as: Using Eq. (10) and (12), one can get The following is the triceps and biceps PAM pressure: where 0 , 0 represent primary pressure of triceps and biceps, respectively, ∆ is designated as system's control input, and it displays how much pressure exists between both the triceps and biceps. Then combined the Eq. (14) and (16), to produce: Eq. (16) could be expressed more succinctly as follows: Where is ( ,) and ( ,) are described by where = 1, 2, … , 6. The classification of coefficients' factors , and have been listed in Table 2. The difference between the pressures is the ∆P and given in Eq. (14) is characterized as a control signal; that is = ∆ . additionally, if state variable 1 is assigned to angular position and state variable 2 denotes angular velocity , then the following describes a state space representation: Eq.  Fig. 4 presents the MATLAB/SIMULINK /R2019a for the PAM actuated arm. The simulation of PAM actuated arm; the model representation is by using Equation (20). Table 3 displays the values of the PAM model's actuated arm variables that were used in the simulations.   The system as well as controller have both been modelled using the MATLAB/SIMULINK software suite. The outputs of the open loop position and velocity are shown in Fig. 5. The PAM's motion is the main issue since it is unstable and unmanageable due to the lack of speed control, which in turn results in undesirable movement that needs to be regulated. Fig. 5. shows that the open loop system is unstable. In order to stabilize the PAM and move its states to the equilibrium point area, the Backstepping controller is used.

III. BACKSTEPPING CONTROL DESIGN (BSC)
The control methods for analyzing the control design for the movement of the PAM robot arm have been developed in this section. A BSC approach is used to develop the control design [28] [29]. The backstepping controller's design variables directly affect how dynamically responsive the controlled system is [30]. Follow the steps mentioned to establish the BSC algorithm for a Single Arm PAM-Actuated Robot system [31].
Let the variation between actual angle position 1 = and needed trajectory 1 = be the as the follow [30], [32][33] is: The error's time derivative, can be written as follows in Eq. (21): Defining the first virtual control 1 = 2 and sub in Eq. (22) to get: The Lyapunov function is a positive function and the function derivative [ This implies 1 < 0 Let the error 2 , between actual state x 2 and the first virtual control 1 described by taking time derivative of Eq. (28) and utilizing Eq. (20) to get: The second Lyapunov function is: Utilizing time derivative of Lyapunov function 2 = 1 1 + 2 2 (32) 2 = − 1 2 1 + 2 ( ( 1 , 2 ) + ( 1 , 2 ) − 1 ) (33) Choosing the control law: The result of the Lyapunov function's derivative is: where, 1 and 2 represent positive constant that is to be determined with the use of the Bat algorithm and 2 < 0 are negative definite [35] [36].

A. Optimal Backstepping Control Parameters
All control systems must operate accurately in both steady-state and transient conditions with a low error rate. there are plenty of methods to optimize the results and improve the control to get better outcomes such as particle swarm optimization (PSO) [37], chaotic particle swarm optimization (CPSO) [38], cuckoo search optimization (CSO) [39], modified chaotic invasive weed optimization (MCIWO) [40]and bat algorithm [41]. In this paper using bat algorithm to optimize the model control. The output and stability of the system are affected by the backstepping algorithm's parameters The objective of the present study is to select the optimal parameters control value for the PAM-Actuated robot arm. These two design parameters are referred to as 1 and 2 . BAT is modeled after how common bats use active sonar to determine where to find food [42]. The Bat method is a popular met heuristic algorithm for resolving practical optimization issues. The Bat must be used by three principles: To begin started, all bats utilize the echolocation in order to measure their distance from a given spot. Second, bats fly in a predetermined direction at a predetermined speed at irregular intervals with a fixed frequency [43]. The volume and wavelength, however, can change. As a result, Bats immediately change their wave-lengths to match their target. Thirdly, the authors believed that the best approach to varying volume is to go from the loudest to the quietest [23]. Different algorithms can be developed inspired by bats, or called bat algorithms. For simplicity, now Using the following rough or idealistic guidelines: 1) All the bats utilize echolocation to determine the range and somehow magically distinguish between background obstacles and food/prey 2) In order to find prey, bats fly at random with a velocity of , a position of , a fixed frequency of , a variable wave-length, and a loudness of 0 . Depending on how close their target is, they can automatically modify pulses' wave-length (or frequency) and rate of emission (element ∈ [0, 1]).
3) Despite the fact that there are numerous ways in which the loudness can change, we suppose that it changed from a high (positive) to a minimal constant value [44].
Additionally, to the previous principles, frequencies and wavelengths are typically set so that they closely reflect the size of the zone of interest. In practical applications, frequency and wavelengths occur within the ranges of here, the ∈ [0-1] represents random vector that has been taken from uniformly distributed; * represents current global best location (i.e., solution) as determined through the comparison of all of the solutions for all of the bats.
A new solution is represented locally for each bat using random walk once a solution has been selected from among actually better possibilities for local research.
In the case where the bat locates a prey, the level of the sound drops and the pulse emission rate rises. The bat is heading to optimum solution based on: here, and are constants ( = = 0.90), the initial emission rate is ∈ [0 − 1], and initial loudness is The [ 1 , 2 ] variables of the suggested controller for a PAM-Actuated robot arm are tuned using BAT methods. The BSC determines and sets the optimized design variable at the algorithm's conclusion. The BA has set the size of the population at 40 and the maximum number of iterations at 100. Mean Square Error (MSE), which may be calculated as follows, is chosen as the cost function which will be utilized in order to assess every one of the particles during the search for minimum.
where, 1 = 1 − 1 , represents sampling number [ 45 ] .  Finally, the BSC sets those optimum values so as to achieve a system that is controlled by the optimal BSC.

V. SIMULATION RESULTS AND DISCUSSION
In this section describes a BSC designed for single-arm PAM-actuated robot stabilization, tracking, and regulatory control. Using MATLAB/SIMULINK/2019a simulation to analyze the performance of the BSC and evaluate the controller. The coefficient values affecting the system for a single-arm PAM-actuated robot are displayed in Table 3. Backstepping controllers based on the try-and-error approach and the Bat algorithm have been compared in terms of performance using the MSE as a performance measure. Table  5. includes the controller design parameter's optimal and trialand-error values.
The current BSC algorithm was validated against Humaidi et al. [46] A comparison was accomplished using the response of stability the position tracking for the PAM actuated arm moving. As mentioned previously that the BSC algorithm requires validation as well. The BSC scenario investigates the response of stability the position tracking when using the proposed BSC with bat algorithms scheme and optimal classical synergetic controller (CSC). The comparison has been performed with the use of the position tracking in the valve during PAM actuated arm moving. The time response for PAM actuated arm moving was obtained from the optimal classical synergetic controller (CSC) reaching its stable state at 4sec, whereas response as a result of the non-optimal CSC reaches the equilibrium at stable state at 6.5sec. but response of using the optimal BSC reaches its steady at 2.5 sec, while non-optimal reaches its steady-state at 6 sec. Backstepping controllers based on the try-and-error method and the Bat algorithm have been compared in terms of performance using the MSE as a performance measure. Primary parameters determining performance of the PAM actuated arm include arm position and velocity. A series of the isolated time steps at different PAM actuated arm movements are shown in Figs. 8,9,10,11, and 12 to present the performance of the PAM actuated arm under investigation. Fig. 8 shows the position's control signal using the BSC technique, the control signal tracking shows that at time 6 sec reaches its steady-state but the signal has chattering and not smooth. The error between the desired signal and the position with non-optimal BSC is 4 × 10 −5 .  Fig. 9 shows the control signal for the position by applying the BAT algorithm with BSC. According to the figure, the system exhibits excellent convergence and high levels of stability in a limited time the signal reaches its steady-state at 0.5 sec and the chattering is smaller, and the signal is smoother. The error between the desired signal and optimal BSC position signal is 1.376 × 10 −6 .
The position tracking results show how much better the proposed BSC with Bat algorithms method is than the BSC at tracking positions. Furthermore, the use of Bat algorithms minimized the tracking position error and the chattering that happened with the BSC tracking position signal.  Fig. 10 shows how the comparison between the optimal BSC, convolution BSC, and the desired trajectory. Fig. 9. Position trajectory for optimal with BSC.
Figs. 8 and 9 shows how much smoother the position tracking is when using the proposed BSC with bat algorithms scheme than when using the Backstepping control. Additionally, the tracking position error was reduced significantly using bat algorithms.
The BSC maximum tracking error for the position tracking trajectory is 0.4 percent at peak. Therefore, a careful analysis of the data reveals that the ideal BSC method reduces tracking errors for the position by a factor of 0.004%. Additionally, the velocity tracking errors presented in Fig. 10. show how much the tracking performance is improved by employing BSC in combination with bat algorithms. Figs. 11 (a-b) show that bat algorithms have an increased tracking velocity for the desired trajectory.

VI. CONCLUSION
The PAM actuated arm was designed and developed in this research utilizing the Backstepping method, It involves modifying the system's equations to incorporate a new state. It is shown that the control law is in charge of providing precise monitoring of the required trajectory for the movements of the arm manipulator. By simulating the control scheme using MATLAB/2019/b. The Bat method is employed to determine the best design parameters for improved dynamic performance, which eliminates the requirement for a trial-and-error method of fine-tuning the design BSC. The accuracy of the simulation results was confirmed by comparing them to those published in previous works on Adaptive Synergetic Control Design. The comparison revealed that utilizing BSC reduces error and employing optimal BSC increases accuracy. As compared to backstepping control using try-and-error parameters the error is 0.4% however, according to the proposed BSC with BA conclusions, the PAM actuated arm has a position inaccuracy of 0.004%.
The future work in this research will focus on a comparative study that can be achieved by utilizing an adaptive control strategy for the same control, robot configuration, the same conditions and same parameters, in addition to benefiting from the improvement of the bat method on the proposed adaptive control and comparing the results.