A New Adaptive Synergetic Control Design for Single Link Robot Arm Actuated by Pneumatic Muscles

This paper suggests a new control design based on the concept of Synergetic Control theory for controlling a one-link robot arm actuated by Pneumatic artificial muscles (PAMs) in opposing bicep/tricep positions. The synergetic control design is first established based on known system parameters. However, in real PAM-actuated systems, the uncertainties are inherited features in their parameters and hence an adaptive synergetic control algorithm is proposed and synthesized for a PAM-actuated robot arm subjected to perturbation in its parameters. The adaptive synergetic laws are developed to estimate the uncertainties and to guarantee the asymptotic stability of the adaptive synergetic controlled PAM-actuated system. The work has also presented an improvement in the performance of proposed synergetic controllers (classical and adaptive) by applying a modern optimization technique based on Particle Swarm Optimization (PSO) to tune their design parameters towards optimal dynamic performance. The effectiveness of the proposed classical and adaptive synergetic controllers has been verified via computer simulation and it has been shown that the adaptive controller could cope with uncertainties and keep the controlled system stable. The proposed optimal Adaptive Synergetic Controller (ASC) has been validated with a previous adaptive controller with the same robot structure and actuation, and it has been shown that the optimal ASC outperforms its opponent in terms of tracking speed and error.


Introduction
Pneumatic actuators such as cylinders, pneumatic stepper motors, bellows, and pneumatic engines are commonly used to date. Pneumatic Artificial Muscles (PAMs) are one type of pneumatic actuator, which are made mainly of inflatable and flexible membrane that works like inverse bellows; i.e., they contract on inflation. The force generated by PAM actuators does not depend only on pressure, but also on the state of inflation, which adds another source of spring-like behavior. These PAMs, which mimic the animal muscle, are characterized by their light weight, since the membrane forms the core element of these actuators. However, they can transfer the same amount of power as cylinders do, when both actuators have the same volume and pressure ranges [1,2].
PAMs are used in many applications due to their light weight, simple construction and high force/weight ratio, direct connection, easy replacement and safe operation. The PAM actuators found is applied to develop the motion equation with massive links utilizing an experimental approach to approximating the force function, determining the values of certain parameters based on the Adaptive Neural Fuzzy Inference System (ANFIS). The work focused on the model characterization and no control strategies have been addressed.
Enzevaee et al. [21] have presented an Active Force Control (AFC) strategy based on a Fuzzy Logic (FL) controller for tracking control of a single-link robot arm. The simulated and experimental results showed that the proposed control scheme could compensate the subjected disturbances effectively and robustly. The main point which has been reported is that the dynamics of the robot arm were approximated by input-output gain. Moreover, the use of a PID controller as the tracking controller could not compensate the uncertainty in system parameters Hiroki [22] presented a control method for a two joint leg driven by PAM. The control method is based on adjusting the timing of cyclic input signal using a genetic algorithm (GA) according to simple cost functions. It has been shown that the robustness of the Pam-driven leg is limited by adjustment of input timing. Unfortunately, the work did not support a control design under uncertainties, but rather it has addressed the problem of robustness improvement based on adjustment of cyclic input timing.
Caldwell et al. in [23] presented an indirect adaptive controller based on pole-placement control technique for braided pneumatic muscle actuator (PMA). The model of PAM actuator has been identified via input-output data according to the proposed model structure. The work reported low bandwidth of the closed-loop system and hence low dynamic speed of response due to limited structure of the polynomial model.
The works reviewed in the above literature either used intelligent control schemes, based on fuzzy logic, neural networks, optimization-based control, nonlinear control based on sliding mode control, adaptive backstepping control, or hybrid nonlinear control. Moreover, it is worthy to mention that the structure of the PAM-actuated manipulator differs from one study to another. Different from the previous studies, in this paper, an adaptive controller is developed for tracking control of PAM-manipulator based on synergetic control strategy.
The Synergetic Control (SC) theory is based a state-space theory, which is utilized for the design and control of highly complex and connected nonlinear systems. This control strategy could enable the state variables of the system to evolve on invariant manifolds chosen by the designer and to achieve the required performance in spite of the presence of uncertainties and disturbances [24,25]. The design of nonlinear systems based on synergetic control can follow the following general procedure [26,27]: Forming the extended system of differential equations, which reflects different operations such as achieving the set values, coordinating observing, optimization, suppressing the disturbances etc. 2.
Synthesizing "external" controls which ensure a reduction in the extra degrees of freedom of the extended system with respect to the final manifold. The motion of the representing point is described by the equations of the system's "internal" dynamics. 3.
Synthesizing the "internal" controls by means of forming the links between the "internal" coordinates of the system. These links ensure the reaching of the control aim. 4.
The synergetic controller directs the trajectories of the system to move onto the manifold from any initial points to their corresponding equilibrium points.
It has been shown that the design parameters of different adaptive and non-adaptive controllers have a direct impact on their performances. These design parameters are often selected based on the trial-and-error procedure. This old technique could not find the optimal dynamic performance of the controlled system based on the proposed controllers. Therefore, a modern optimization technique is used to tune these design parameters to improve the dynamic performance of the controlled system [28,29]. In the present work, the Particle Swarm Optimization (PSO) has been suggested to adjust the design parameters of the controlled system. This modern optimization technique was first proposed by Kennedy in 1995 and it was inspired by the behavior of organisms [30]. This optimization tuner is characterized by fast convergence, the efficiency of computation and it has the capability to find local and global solutions [31,32]. Other modern and generalized optimization techniques can be employed either to improve the optimization process or to make a comparison in performance among each other [33][34][35][36][37][38][39][40][41].
The main problem with systems actuated by Pneumatic Artificial Muscles is that they suffer from external disturbances, uncertainties (e.g., unknown parameters and unmodeled uncertainties, etc.), high nonlinearities, hysteresis and time varying characteristics, which dramatically degrade the performance of tracking control. To handle this important issue, an adaptive control method is proposed for the PAM-actuated one-link robot arm, which can deal effectively with the influence produced by parametric uncertainties in actuating muscles.
The contribution of this paper is to develop an adaptive control algorithm based on synergetic control theory for tracking control of a single-link robot arm actuated by pneumatic artificial muscles under uncertainties in muscle parameters. The proposed controller is continuous and able to prevent chattering and it can simultaneously compensate the parametric uncertainties, and it yields bounded adaptive gains. The improvement in performance of the proposed adaptive controller by PSO-tuning of its design parameters is the second contribution, which can be added as well since it has never been addressed in the literature for such PAM systems. The rfirst controller is based on known parameters of PAM-actuated one-link robot arms named as classical synergetic controller (CSC), while the other adaptive synergetic controller (ASC) solves the problem of unknown uncertainties in system parameters. The stability of the controlled PAM robotic system is analyzed and proved based on Lypunov stability analysis. In addition, the PSO technique is introduced for tuning the designed parameters of proposed controllers to better enhance the performances of the proposed controllers. Thus, the contribution of the present work can be embodied by pursuing the following steps of objectives: by Kennedy in 1995 and it was inspired by the behavior of organisms [30]. This optimization tuner is characterized by fast convergence, the efficiency of computation and it has the capability to find local and global solutions [31,32]. Other modern and generalized optimization techniques can be employed either to improve the optimization process or to make a comparison in performance among each other [33][34][35][36][37][38][39][40][41].
The main problem with systems actuated by Pneumatic Artificial Muscles is that they suffer from external disturbances, uncertainties (e.g., unknown parameters and unmodeled uncertainties, etc.), high nonlinearities, hysteresis and time varying characteristics, which dramatically degrade the performance of tracking control. To handle this important issue, an adaptive control method is proposed for the PAM-actuated one-link robot arm, which can deal effectively with the influence produced by parametric uncertainties in actuating muscles.
The contribution of this paper is to develop an adaptive control algorithm based on synergetic control theory for tracking control of a single-link robot arm actuated by pneumatic artificial muscles under uncertainties in muscle parameters. The proposed controller is continuous and able to prevent chattering and it can simultaneously compensate the parametric uncertainties, and it yields bounded adaptive gains. The improvement in performance of the proposed adaptive controller by PSO-tuning of its design parameters is the second contribution, which can be added as well since it has never been addressed in the literature for such PAM systems. The first controller is based on known parameters of PAM-actuated one-link robot arms named as classical synergetic controller (CSC), while the other adaptive synergetic controller (ASC) solves the problem of unknown uncertainties in system parameters. The stability of the controlled PAM robotic system is analyzed and proved based on Lypunov stability analysis. In addition, the PSO technique is introduced for tuning the designed parameters of proposed controllers to better enhance the performances of the proposed controllers. Thus, the contribution of the present work can be embodied by pursuing the following steps of objectives:  To design a new classical synergetic control (CSC) algorithm for the PAM-actuated robot arm based on Lypunove-based stability analysis.  To design a new adaptive synergetic control algorithm to cope with the problem of uncertainties inherited in parameters of the PAM-actuated robot arm.  To prove the asymptotic stability of the PAM-based robot arm controlled by CSC and ASC, such that all errors finally converge to their corresponding zero equilibrium points based on Lypunove stability analysis.  To better improve the dynamic performance of the PAM-actuated robot arm controlled by proposed controllers by replacing the trial-and-error procedure with the PSO technique for optimal tuning of controllers' design parameters towards better performance of controllers.

The Dynamic Model of the PAM-Actuated Single-Link Robot Arm
Before proceeding in the control design of the PAM system, it is important to first develop the mathematical model for the system, which mimicks the actual behavior of real system. Then, the system can be analyzed and its associated controller can be designed to meet the required performance. The single link robotic arm is shown in Figure 1 [12].
To design a new classical synergetic control (CSC) algorithm for the PAM-actuated robot arm based on Lypunove-based stability analysis. by Kennedy in 1995 and it was inspired by the behavior of organisms [30]. This optimization tuner is characterized by fast convergence, the efficiency of computation and it has the capability to find local and global solutions [31,32]. Other modern and generalized optimization techniques can be employed either to improve the optimization process or to make a comparison in performance among each other [33][34][35][36][37][38][39][40][41].
The main problem with systems actuated by Pneumatic Artificial Muscles is that they suffer from external disturbances, uncertainties (e.g., unknown parameters and unmodeled uncertainties, etc.), high nonlinearities, hysteresis and time varying characteristics, which dramatically degrade the performance of tracking control. To handle this important issue, an adaptive control method is proposed for the PAM-actuated one-link robot arm, which can deal effectively with the influence produced by parametric uncertainties in actuating muscles.
The contribution of this paper is to develop an adaptive control algorithm based on synergetic control theory for tracking control of a single-link robot arm actuated by pneumatic artificial muscles under uncertainties in muscle parameters. The proposed controller is continuous and able to prevent chattering and it can simultaneously compensate the parametric uncertainties, and it yields bounded adaptive gains. The improvement in performance of the proposed adaptive controller by PSO-tuning of its design parameters is the second contribution, which can be added as well since it has never been addressed in the literature for such PAM systems. The first controller is based on known parameters of PAM-actuated one-link robot arms named as classical synergetic controller (CSC), while the other adaptive synergetic controller (ASC) solves the problem of unknown uncertainties in system parameters. The stability of the controlled PAM robotic system is analyzed and proved based on Lypunov stability analysis. In addition, the PSO technique is introduced for tuning the designed parameters of proposed controllers to better enhance the performances of the proposed controllers. Thus, the contribution of the present work can be embodied by pursuing the following steps of objectives:  To design a new classical synergetic control (CSC) algorithm for the PAM-actuated robot arm based on Lypunove-based stability analysis.  To design a new adaptive synergetic control algorithm to cope with the problem of uncertainties inherited in parameters of the PAM-actuated robot arm.  To prove the asymptotic stability of the PAM-based robot arm controlled by CSC and ASC, such that all errors finally converge to their corresponding zero equilibrium points based on Lypunove stability analysis.  To better improve the dynamic performance of the PAM-actuated robot arm controlled by proposed controllers by replacing the trial-and-error procedure with the PSO technique for optimal tuning of controllers' design parameters towards better performance of controllers.

The Dynamic Model of the PAM-Actuated Single-Link Robot Arm
Before proceeding in the control design of the PAM system, it is important to first develop the mathematical model for the system, which mimicks the actual behavior of real system. Then, the system can be analyzed and its associated controller can be designed to meet the required performance. The single link robotic arm is shown in Figure 1 [12].
To design a new adaptive synergetic control algorithm to cope with the problem of uncertainties inherited in parameters of the PAM-actuated robot arm. by Kennedy in 1995 and it was inspired by the behavior of organisms [30]. This optimization tuner is characterized by fast convergence, the efficiency of computation and it has the capability to find local and global solutions [31,32]. Other modern and generalized optimization techniques can be employed either to improve the optimization process or to make a comparison in performance among each other [33][34][35][36][37][38][39][40][41].
The main problem with systems actuated by Pneumatic Artificial Muscles is that they suffer from external disturbances, uncertainties (e.g., unknown parameters and unmodeled uncertainties, etc.), high nonlinearities, hysteresis and time varying characteristics, which dramatically degrade the performance of tracking control. To handle this important issue, an adaptive control method is proposed for the PAM-actuated one-link robot arm, which can deal effectively with the influence produced by parametric uncertainties in actuating muscles.
The contribution of this paper is to develop an adaptive control algorithm based on synergetic control theory for tracking control of a single-link robot arm actuated by pneumatic artificial muscles under uncertainties in muscle parameters. The proposed controller is continuous and able to prevent chattering and it can simultaneously compensate the parametric uncertainties, and it yields bounded adaptive gains. The improvement in performance of the proposed adaptive controller by PSO-tuning of its design parameters is the second contribution, which can be added as well since it has never been addressed in the literature for such PAM systems. The first controller is based on known parameters of PAM-actuated one-link robot arms named as classical synergetic controller (CSC), while the other adaptive synergetic controller (ASC) solves the problem of unknown uncertainties in system parameters. The stability of the controlled PAM robotic system is analyzed and proved based on Lypunov stability analysis. In addition, the PSO technique is introduced for tuning the designed parameters of proposed controllers to better enhance the performances of the proposed controllers. Thus, the contribution of the present work can be embodied by pursuing the following steps of objectives:  To design a new classical synergetic control (CSC) algorithm for the PAM-actuated robot arm based on Lypunove-based stability analysis.  To design a new adaptive synergetic control algorithm to cope with the problem of uncertainties inherited in parameters of the PAM-actuated robot arm.  To prove the asymptotic stability of the PAM-based robot arm controlled by CSC and ASC, such that all errors finally converge to their corresponding zero equilibrium points based on Lypunove stability analysis.  To better improve the dynamic performance of the PAM-actuated robot arm controlled by proposed controllers by replacing the trial-and-error procedure with the PSO technique for optimal tuning of controllers' design parameters towards better performance of controllers.

The Dynamic Model of the PAM-Actuated Single-Link Robot Arm
Before proceeding in the control design of the PAM system, it is important to first develop the mathematical model for the system, which mimicks the actual behavior of real system. Then, the system can be analyzed and its associated controller can be designed to meet the required performance. The single link robotic arm is shown in Figure 1 [12].
To prove the asymptotic stability of the PAM-based robot arm controlled by CSC and ASC, such that all errors finally converge to their corresponding zero equilibrium points based on Lypunove stability analysis. by Kennedy in 1995 and it was inspired by the behavior of organisms [30]. This optimization tuner is characterized by fast convergence, the efficiency of computation and it has the capability to find local and global solutions [31,32]. Other modern and generalized optimization techniques can be employed either to improve the optimization process or to make a comparison in performance among each other [33][34][35][36][37][38][39][40][41].
The main problem with systems actuated by Pneumatic Artificial Muscles is that they suffer from external disturbances, uncertainties (e.g., unknown parameters and unmodeled uncertainties, etc.), high nonlinearities, hysteresis and time varying characteristics, which dramatically degrade the performance of tracking control. To handle this important issue, an adaptive control method is proposed for the PAM-actuated one-link robot arm, which can deal effectively with the influence produced by parametric uncertainties in actuating muscles.
The contribution of this paper is to develop an adaptive control algorithm based on synergetic control theory for tracking control of a single-link robot arm actuated by pneumatic artificial muscles under uncertainties in muscle parameters. The proposed controller is continuous and able to prevent chattering and it can simultaneously compensate the parametric uncertainties, and it yields bounded adaptive gains. The improvement in performance of the proposed adaptive controller by PSO-tuning of its design parameters is the second contribution, which can be added as well since it has never been addressed in the literature for such PAM systems. The first controller is based on known parameters of PAM-actuated one-link robot arms named as classical synergetic controller (CSC), while the other adaptive synergetic controller (ASC) solves the problem of unknown uncertainties in system parameters. The stability of the controlled PAM robotic system is analyzed and proved based on Lypunov stability analysis. In addition, the PSO technique is introduced for tuning the designed parameters of proposed controllers to better enhance the performances of the proposed controllers. Thus, the contribution of the present work can be embodied by pursuing the following steps of objectives:  To design a new classical synergetic control (CSC) algorithm for the PAM-actuated robot arm based on Lypunove-based stability analysis.  To design a new adaptive synergetic control algorithm to cope with the problem of uncertainties inherited in parameters of the PAM-actuated robot arm.  To prove the asymptotic stability of the PAM-based robot arm controlled by CSC and ASC, such that all errors finally converge to their corresponding zero equilibrium points based on Lypunove stability analysis.  To better improve the dynamic performance of the PAM-actuated robot arm controlled by proposed controllers by replacing the trial-and-error procedure with the PSO technique for optimal tuning of controllers' design parameters towards better performance of controllers.

The Dynamic Model of the PAM-Actuated Single-Link Robot Arm
Before proceeding in the control design of the PAM system, it is important to first develop the mathematical model for the system, which mimicks the actual behavior of real system. Then, the system can be analyzed and its associated controller can be designed to meet the required performance. The single link robotic arm is shown in Figure 1 [12].
To better improve the dynamic performance of the PAM-actuated robot arm controlled by proposed controllers by replacing the trial-and-error procedure with the PSO technique for optimal tuning of controllers' design parameters towards better performance of controllers.

The Dynamic Model of the PAM-Actuated Single-Link Robot Arm
Before proceeding in the control design of the PAM system, it is important to first develop the mathematical model for the system, which mimicks the actual behavior of real system. Then, the system can be analyzed and its associated controller can be designed to meet the required performance. The single link robotic arm is shown in Figure 1 [12]. The amount of pneumatic muscle extension and muscle contraction can be expressed respectively by [12,16], The forearm forms an angle = ( ⁄ ) with the tricep cable. The forearm is permitted to rotate within this angle. The angle = 0 corresponds to the case that the forearm is in a downward position, while the angle = represents the case that the forearm is in an extreme upward position. The clockwise torque generated by the bicep muscle on the fore arm is given by: The counterclockwise torque applied by the tricep muscle is described by: where (. ) and (. ) are the developed forces from the tricep and bicep PAMs, respectively, is the pulley radius. The generated (. ) and (. ) can be described by the following dynamic PAM model [12]: where ( ) , ( ) and ( ) are the bicep PAM force, spring and viscosity coefficients, respectively, and they are expressed as follows: In addition, ( ) , ( ) and ( ) represent the tricep PAM force, spring and viscosity coefficients, respectively, and they are defined by the following expressions: where, m is the mass (kg), g is the gravitational acceleration (m/s 2 ), B(P b ) is the bicep coefficient of viscous friction, L is the length of the arm from the center of the mass to the joint, B(P t ) is the tricep coefficient of viscous friction, K(P b ) represents the bicep spring coefficient (N/m), K(P t ) represents the tricep spring coefficient (N/m). F(P b ) is the force exerted by PAM in the bicep case, F(P t ) is the force exerted by PAM in the tricep case, a is the distance from the joint axis of rotation to the PMs attached point (A), r is the pulley radius. The amount of pneumatic muscle extension x t and muscle contraction x b can be expressed respectively by [12,16], The forearm forms an angle α = sin −1 (r/a) with the tricep cable. The forearm is permitted to rotate within this angle. The angle θ = 0 corresponds to the case that the forearm is in a downward position, while the angle θ = π represents the case that the forearm is in an extreme upward position. The clockwise torque generated by the bicep muscle on the fore arm is given by: The counterclockwise torque applied by the tricep muscle is described by: where F t (.) and F b (.) are the developed forces from the tricep and bicep PAMs, respectively, r is the pulley radius. The generated F t (.) and F b (.) can be described by the following dynamic PAM model [12]: where F(P b ), K(P b ) and B(P b ) are the bicep PAM force, spring and viscosity coefficients, respectively, and they are expressed as follows: In addition, F(P t ), K(P t ) and B(P t ) represent the tricep PAM force, spring and viscosity coefficients, respectively, and they are defined by the following expressions: It is worthy to note that the coefficient B depends on wether the muscle is being deflated or inflated; that is, one has to differentiate between the tricep and bicep coefficients B(P t ) and B(P b ).
The dynamic equation of motion can be found by summing the torques, described by Equations (3) and (4), about the elbow I ..
where I = ML 2 represents the moment of mass inertia about the elbow and the last term (MgLsinθ) has been added to account for the counterclockwise torque exerted on the forearm due to the mass gravity. Substituting Equations (5) and (6) into Equation (9), one can obtain: The time derivative of PM extension x t and contraction x b are given, respectively, as: Using Equations (10)- (12), one can get The pressure of tricep and bicep PAM is given where P 0t , P 0b are the initial pressure of the tricep and bicep, respectively, ∆P is assigned as the control input of the system and it describes the pressure difference between the tricep and bicep. Combining Equations (14)- (16), to yield I. ..
Equation (16) can be rewritten in the following compact form: ..
where f θ, θ where i = 1, 2, · · · , 6. The definitions of the elements of the coefficients f i , Z i and b i are listed in Table 1.
The variation in the pressure ∆P given in Equation (14) is considered as the control signal; that is u = ∆P. In addition, if the state variable x 1 is assigned to the angular position θ and the state variable x 2 represents the angular position velocity . θ, then state space representation can be described by .

Classical and Adaptive Synergetic Control Design for Single Arm PAM-Actuated Robot
In this section, two control schemes have been developed for tracking control design of angular position for PAM robot arm. The first control design is established based on the classical synergetic control method. The second control design presents the development of an adaptive synergetic control algorithm for tracking control of the PAM robot arm subjected to uncertainties in system parameters.

Synergetic Control Design
Let e be the difference between the actual angle position x 1 = θ and the desired trajectory x 1d = θ d as follows: Using dynamic equation of the PAM robot arm, the first and second derivatives of error equations are given by, respectively, .. e = .
x 2 − .. x where u is the control signal.
where c c is a scalar design parameter concerning CSC strategy. In order to ensure the stability and to guarantee the convergence of the state trajectories to their corresponding desired manifolds and remain on it for future time, the dynamic evolution of the macro-variable towards the manifolds is defined as follows: where T > 0 represents the rate of convergence of the macro-variable to manifolds ψ(e) = 0. Using Equations (23) and (24) becomes Using the dynamic equation of the PAM robot arm, Equation (25) becomes Using Equation (21), one can have In order to ensure the dynamic T . ψ(x) + ψ( x) = 0, the control law u can be designed as Proof. The candidate Lypunov function is chosen in terms of micro-variables vector as follows: Taking The time derivative of Equation (29) gives Using Equations (24) and (30) becomes This indicates that the control law of Equation (28) guarantees the system stability of the PAM robotic System. Theorem 1. For the dynamic equation of the PAM-actuated robot arm, the control signal u can be acquired by Equation (28), which ensures the stability of system motion and stays in the desired manifold.

Design of Adaptive Synergetic Control for Single-Link Robot Arm
In order to address the problem of uncertainty, which is a characteristic feature in the physical parameters of the PAMs system, and for suppressing the effects of undesired disturbances that may have an effect on the tracking performance, the ASC has been introduced to develop the necessary adaptive laws that can estimate these uncertain parameters such that the asymptotic stability of the adaptive controlled system is guaranteed. In this case, the adaptive syngeneic controller could direct all trajectories of micro-variables to the equilibrium or manifold asymptotically. When the trajectory reaches the manifold, the synergetic controller will maintain it there thereafter. Assumption 1. Two coefficients are permitted to be uncertain in their values; namely, the viscous friction coefficient and spring coefficient.
The variation in coefficients of bicep muscle has only been taken into account, that is, the uncertainty in viscosity coefficient B ob and the uncertainty in spring coefficient K ob . However, since the spring coefficient in the bicep and tricep muscles are the same, then K ob is assigned to the coefficient K o . Assumption 3. The inflation case will be only considered in developing the control law of the ASC algorithm. (8) and (10), there are three possible parameters of the system, which permit variations in their values; namely, F(p), K(p) and B(p). The assumption 1 indicates that the present work address only the uncertainty in K(p) and B(p), since they are highly subjected to variation during the work of the actuating muscles. According to the above assumptions, the adaptive control law of ASC for the PAM-actuated robot arm will be established based on Lypunov stability analysis. LetB 0b andK 0 denote the estimated values of the viscous friction coefficient and spring coefficient, respectively, which are given byB

Referring to Equations
where,B 0b ,K 0 represent the estimated value of B 0b and K 0 , B 0b and K 0 represent the nominal values and B 0b , K 0 are the variation in B, K, respectively. In order to derive the adaptation law, the candidate L.F. has been chosen in terms of estimation errors as follows: where, γ 1 , γ 2 are the adaptation gains. The first time derivative of Equation (34) is given by Based on Equation (23), one can get where c a is a positive scalar constant concerning the design of the adaptive synergetic controller. Using Equation (21) to have The ideal control law, given in Equation (28), is no longer applicable in the presence of uncertainty and the actual control law is defined in terms of estimated functionf rather than its ideal one as follows; Substituting Equation (38) in Equation (37) to become or, Since, and,f By subtracting f −f and substituting the result in Equation (37), one can have In order to ensure . V ≤ 0, the following terms are enforced to be set to zero; that is, Accordingly, the following adaptation laws can be deduced Using Equations (40)-(42) becomes Since the time derivative of L.F ( . V) is definitely negative, therefore, the proposed ABSMC can guarantee the asymptotic stability of the system even with the presence of uncertainties in parameters (B 0b and K 0 ) of the PAM-actuated robot arm. Theorem 2. If the dynamic system of the PAM-actuated manipulator is subjected to uncertainty in viscosity coefficient B ob and spring coefficient K ob , the developed adaptive laws based on synergetic control methodology described by Equations (44)

Improvement of Controllers' Performances Based on PSO Technique
In order to improve the performances of the proposed controllers (CMC and ASC), the design parameters of these controllers have to be tuned to towards better performance of the controlled system. Trial-and-error procedure for finding or setting these design parameters is cumbersome and it does not lead to an optimal solution in terms of better dynamic performance of controlled systems. As such, the PSO technique has been suggested to find the optimal values of design parameters. It is

Improvement of Controllers' Performances Based on PSO Technique
In order to improve the performances of the proposed controllers (CMC and ASC), the design parameters of these controllers have to be tuned to towards better performance of the controlled system. Trial-and-error procedure for finding or setting these design parameters is cumbersome and it does not lead to an optimal solution in terms of better dynamic performance of controlled systems. As such, the PSO technique has been suggested to find the optimal values of design parameters. It is well-known that the PSO algorithm has the capability to guide the performance of controllers towards optimality. In the case of CSC, the PSO technique is responsible for tuning the design parameter c c , while it undertakes the tuning of constant c a in the case of ASC.
In PSO, each particle navigates around the search (solution) space by updating their velocity according to its own and according to searching experience of other particles. Each particle must update its velocity and position according to the number of iterations, which will be performed according to some cost function to minimize or maximize in each case. In our design, the cost function has to be minimized. The velocity of each particle is updated according to the following equation [42,43]: where, w represents the inertia coefficient, C 1 represents the personal acceleration coefficient and C 2 represents the social acceleration coefficient. The position of each particle is updated by the equation [43,44]: where X k i and X k+1 i represents the current and updated vectors, respectively. The cost function used to evaluate each particle during the search of the minimum is chosen to be the Root Mean Square Error f (RMSE) function.
The performance of the PSO algorithm is sensitive to PSO parameters. The adjusting of inertia value w strikes a better balance between global exploration and local exploitation. The population size is set to compromise between convergence rate and computation time. Acceleration constants C 1 and C 2 represent the particle stochastic acceleration weight toward the personal best (p best ) and the global best (g best ). The effect of their setting is either inducing the particle wandering away in the goal area or moving quickly to the goal area. The tuning of PSO parameters can also be performed either using another overlaying optimizer, a concept known as meta-optimization, or even fine-tuned during the optimization. The latter tuning procedure has been adopted in the present work, where the parameters have been tuned for various optimization scenarios as indicated in Table 2. Table 2. List of parameters assigned to the Particle Swarm Optimization (PSO) algorithm.

Parameters of PSO Technique Value
The inertia coefficient w 1.4 The personal acceleration coefficient C 1 2 The social acceleration coefficient C 2 2 The swarm size (population size) 30 The number of iteration 300

Computer Simulation
The numerical values of the pair of PAM-actuated robot arms in the bicep/tricep positions are listed in Table 3. By using the MATLAB/SIMULINK software package, the PAM-actuated robot, based on proposed controllers, has been modeled and coded in Matlab m-functions and Simulink library blocks as shown in Figure 3. Table 3. Numerical values of system parameters.

Coefficient Description Value
Nominal force exerted by PAM F 0 0.986 × 10 2 N Variation in force exerted by PAM The numerical values of the pair of PAM-actuated robot arms in the bicep/tricep positions are listed in Table 3. By using the MATLAB/SIMULINK software package, the PAM-actuated robot, based on proposed controllers, has been modeled and coded in Matlab m-functions and Simulink library blocks as shown in Figure 3.   The simulink model of the adaptive synegetic controlled robotic system tries to mimick the block diagram of Figure 2. The model consists of three blocks of matlab functions and two subsystems. The blocks of matlab functions are assigned to the robotic system, adaptive law and control law. The algorithms for control law and adaptive law are written in Matlab code and they are brought to the Simulink environment by the matalb function blocks, which can be drawn by the simulink library. The subsystems used in the simulink model are devoted either to generate trajectory signals or to synthesize the micro-variables of the synergetic controller. The simulation parameters used in the present work include a stop time (60 s), a solver type (ode45) with a variable step based on min step size (1 × 10 −5 s) and max step size (1 × 10 −3 s). (One can visit the link cited by Reference [45] for a detailed description of the program source to establish the next scenarios).
The design parameters for the CSC are c c and T, while those for the ASC are also c a and T. The value of parameter T for CSC is set equal to unity, and that for ASC is assumed to be 0.1. The adaption gain γ 1 in the case of ASC based on bicep coefficient of viscosity B 0b has been fixed at value γ 1 = 1 × 10 −9 , while adaption gain γ 2 in the case of uncertainty in spring coefficient K 0 is set with value γ 2 = 1 × 10 −10 . In order to reach an improved dynamic performance of the controlled PAM-actuated robot arm, the PSO algorithm has been incorporated to tune the design parameters of CSC and ASC. Figure 4 shows the open-loop response for the PAM-actuated single-arm robot tested by sin-wave input of unity amplitude and frequency (1 rad/s). synthesize the micro-variables of the synergetic controller. The simulation parameters used in the present work include a stop time (60 s), a solver type (ode45) with a variable step based on min step size (1 × 10 s) and max step size (1 × 10 s). (One can visit the link cited by Reference [45] for a detailed description of the program source to establish the next scenarios). Figure 4 shows the open-loop response for the PAM-actuated single-arm robot tested by sinwave input of unity amplitude and frequency (1 rad/s). The design parameters for the CSC are and , while those for the ASC are also and . The value of parameter for CSC is set equal to unity, and that for ASC is assumed to be 0.1. The adaption gain in the case of ASC based on bicep coefficient of viscosity has been fixed at value = 1 × 10 , while adaption gain in the case of uncertainty in spring coefficient is set with value = 1 × 10 . In order to reach an improved dynamic performance of the controlled PAM-actuated robot arm, the PSO algorithm has been incorporated to tune the design parameters of CSC and ASC. The Root Mean Square Error (RMSE) is the fitness function that has been used to evaluate the iterative particles within the PSO algorithm for all controllers. Figures 5 and 6 show the behavior of cost function with respect to iteration for the controlled PAM single-arm robot based on CSC and ASC, respectively. It is evident that the PSO could successfully guide the design parameters of the proposed controllers to their best solution. However, the PSO algorithm for ASC shows faster convergence than CSC. The convergence of cost function is reached after 15 iterations from the start of the algorithm in the case of CSC, while the cost function finds its global minimum in less than 5 iterations in the case of ASC. The optimal values of tuned parameters will be ready at the end of the algorithm. The controller is termed as "optimal controller" when it is assigned their optimal tuned parameters. Table 4 gives the set of optimal design parameters based on the PSO technique for the PAMactuated robot system controlled by CSC and ASC. The other set of design parameters has been chosen on a trial-and-error basis. The Root Mean Square Error (RMSE) is the fitness function that has been used to evaluate the iterative particles within the PSO algorithm for all controllers. Figures 5 and 6 show the behavior of cost function with respect to iteration for the controlled PAM single-arm robot based on CSC and ASC, respectively. It is evident that the PSO could successfully guide the design parameters of the proposed controllers to their best solution. However, the PSO algorithm for ASC shows faster convergence than CSC. The convergence of cost function is reached after 15 iterations from the start of the algorithm in the case of CSC, while the cost function finds its global minimum in less than 5 iterations in the case of ASC. The optimal values of tuned parameters will be ready at the end of the algorithm. The controller is termed as "optimal controller" when it is assigned their optimal tuned parameters.        Table 4 gives the set of optimal design parameters based on the PSO technique for the PAM-actuated robot system controlled by CSC and ASC. The other set of design parameters has been chosen on a trial-and-error basis. Three scenarios are presented in the present work. In the first scenario, the uncertainty is not taken into account, the second scenario considered the uncertainty condition, while the third scenario has been devoted to the validation purpose. The desired angular positions for the first and second scenarios are defined by where, f 1 = 0.02 Hz, f 2 = 0.05 Hz and f 3 = 0.09 Hz. Figure 7 shows the tracking performance of the joint angle x 1 to the desired trajectory based on optimal and non-optimal CSC. According to the Figure 6, the response due to optimal controller shows faster tracking control than that based on non-optimal controller, where the response of optimal CSC reaches its steady-state at 4 s, while the response due to non-optimal CSC reaches its equilibrium at steady state at 6.5 s. The tracking errors (e) for the PAM-actuated robot arm based on optimal and non-optimal CSCs are shown in Figure 8. It is clear from the figure that the optimal CSC gives better performance than the non-optimal version in terms of tracking errors However, to evaluate this one can easily compute the variances of errors resulting from each controller. It has been shown that the RMSE value based on the optimal controller is given by 0.1686 rad, while that based on the non-optimal one is equal to 0.2563 rad. This indicates that the PSO grants the optimal CSC better tracking error performance than its opponent. Figure 9 illustrates the behaviors of angular velocities of the controlled system based on the PSO algorithm and trial-and-error procedure. The velocity of non-optimal controller has higher peak response (1.62 rad/s) than that based on a trial-and-error basis (1.1 rad/s). Therefore, one can conclude that the dynamic response obtained by optimal CSC outperforms that based on a trial-and-error procedure in terms of transient characteristics. gives better performance than the non-optimal version in terms of tracking errors However, to evaluate this one can easily compute the variances of errors resulting from each controller. It has been shown that the RMSE value based on the optimal controller is given by 0.1686 rad, while that based on the non-optimal one is equal to 0.2563 rad. This indicates that the PSO grants the optimal CSC better tracking error performance than its opponent. Figure 9 illustrates the behaviors of angular velocities of the controlled system based on the PSO algorithm and trial-and-error procedure. The velocity of non-optimal controller has higher peak response (1.62 rad/s) than that based on a trialand-error basis (1.1 rad/s). Therefore, one can conclude that the dynamic response obtained by optimal CSC outperforms that based on a trial-and-error procedure in terms of transient characteristics.   Tracking performance for the PAM actuated robot arm with optimal and non-optimal CSC.

Scenario I: PAM-actuated Robot Arm based on CSC
. Figure 8. Tracking Error of PAM system controlled by optimal and non-optimal CSCs. Figure 8. Tracking Error of PAM system controlled by optimal and non-optimal CSCs.
Entropy 2020, 22, x FOR PEER REVIEW 16 of 24 Figure 9. Behaviors of angular velocities for PAM actuated robot arm system controlled with optimal and non-optimal CSC.
The corresponding control effects based on optimal and non-optimal CSC are depicted in Figure  10. One can evaluate the energy or control signal exerted by both controllers by calculating the RMS of the control signal over the entire simulation time. It has been shown that the RMS value of pressure input (control signal) is equal to (1.1367 × 10 Pa) for the optimal controller, while the value of RMS of the control signal for the non-optimal controller is equal to (1.1095 × 10 Pa). This indicates that the optimal controller takes more energy than that based on the non-optimal one. This is the price paid by the optimal controller to have a better dynamic response. The corresponding control effects based on optimal and non-optimal CSC are depicted in Figure 10. One can evaluate the energy or control signal exerted by both controllers by calculating the RMS of the control signal over the entire simulation time. It has been shown that the RMS value of pressure input (control signal) is equal to 1.1367 × 10 5 Pa for the optimal controller, while the value of RMS of the control signal for the non-optimal controller is equal to (1.1095 × 10 5 Pa). This indicates that the optimal controller takes more energy than that based on the non-optimal one. This is the price paid by the optimal controller to have a better dynamic response.

Scenario II: PAM-actuated Robot Arm Based on ASC
In order to design an adaptive controller (ASC), the bicep coefficient of viscosity B 0b and the spring coefficient K have been considered as uncertainty factors. The designed ASC will develop adaptive laws, which can estimate the uncertain coefficients (B 0b ,K). The linear angle position behavior based on optimal and non-optimal ASC is shown Figure 11. It is clear from zoomed capture of the figure that the optimal ASC has faster tracking performance than that based on non-optimal ASC. This can be proved numerically, where the response due to the optimal controller concedes with the reverence input after 0.6 s from start-up, while the response based on the non-optimal controller will reach the reference trajectory in 1.3 s. 10. One can evaluate the energy or control signal exerted by both controllers by calculating the RMS of the control signal over the entire simulation time. It has been shown that the RMS value of pressure input (control signal) is equal to (1.1367 × 10 Pa) for the optimal controller, while the value of RMS of the control signal for the non-optimal controller is equal to (1.1095 × 10 Pa). This indicates that the optimal controller takes more energy than that based on the non-optimal one. This is the price paid by the optimal controller to have a better dynamic response.

Scenario II: PAM-actuated Robot Arm Based on ASC
In order to design an adaptive controller (ASC), the bicep coefficient of viscosity and the spring coefficient have been considered as uncertainty factors. The designed ASC will develop adaptive laws, which can estimate the uncertain coefficients ( , ). The linear angle position behavior based on optimal and non-optimal ASC is shown Figure 11. It is clear from zoomed capture of the figure that the optimal ASC has faster tracking performance than that based on non-optimal ASC. This can be proved numerically, where the response due to the optimal controller concedes with the reverence input after 0.6 s from start-up, while the response based on the non-optimal controller will reach the reference trajectory in 1.3 s.  Figure 12 shows the tracking error ( ) between the optimal and non-optimal ASC controlled PAM systems. In a numerical sense, the RMSE value resulting from optimal ASC is equal to 0.0483 rad, while the RMSE given by non-optimal ASC is equal to 0.0530 rad. This indicates that the optimal ASC gives better tracking performance and better error variance than that based on trialand-error procedure. Figure 13 illustrates the linear velocity behaviors for both optimal and trial-anderror procedure for ASC. However, one can see that the peak of velocity response is slightly higher than that resulting from its counterpart.  Figure 12 shows the tracking error (e) between the optimal and non-optimal ASC controlled PAM systems. In a numerical sense, the RMSE value resulting from optimal ASC is equal to 0.0483 rad, while the RMSE given by non-optimal ASC is equal to 0.0530 rad. This indicates that the optimal ASC gives better tracking performance and better error variance than that based on trial-and-error procedure. Figure 13 illustrates the linear velocity behaviors for both optimal and trial-and-error procedure for ASC. However, one can see that the peak of velocity response is slightly higher than that resulting from its counterpart. Figure 14 combines the effects of ASC control signals u due to optimal and trial-and-error procedures. It is interesting to calculate the input pressure variances based on the proposed controllers to evaluate the amount of pressure absorbed by the controlled system. The RMS values of control signals over the entire time of simulation resulting from optimal ASC and non-optimal ASC are 7.5494 × 10 6 and 1.2366 × 10 6 , respectively. It is clear that the input pressure taken by optimal ASC is higher than that absorbed by non-optimal controller; this is the price which has to be paid by the optimal controller to have better dynamic and tracking performance. PAM systems. In a numerical sense, the RMSE value resulting from optimal ASC is equal to 0.0483 rad, while the RMSE given by non-optimal ASC is equal to 0.0530 rad. This indicates that the optimal ASC gives better tracking performance and better error variance than that based on trialand-error procedure. Figure 13 illustrates the linear velocity behaviors for both optimal and trial-anderror procedure for ASC. However, one can see that the peak of velocity response is slightly higher than that resulting from its counterpart.   Figure 14 combines the effects of ASC control signals due to optimal and trial-and-error procedures. It is interesting to calculate the input pressure variances based on the proposed controllers to evaluate the amount of pressure absorbed by the controlled system. The RMS values of control signals over the entire time of simulation resulting from optimal ASC and non-optimal ASC are 7.5494 × 10 and 1.2366 × 10 , respectively. It is clear that the input pressure taken by optimal ASC is higher than that absorbed by non-optimal controller; this is the price which has to be paid by the optimal controller to have better dynamic and tracking performance.  Figure 14 combines the effects of ASC control signals due to optimal and trial-and-error procedures. It is interesting to calculate the input pressure variances based on the proposed controllers to evaluate the amount of pressure absorbed by the controlled system. The RMS values of control signals over the entire time of simulation resulting from optimal ASC and non-optimal ASC are 7.5494 × 10 and 1.2366 × 10 , respectively. It is clear that the input pressure taken by optimal ASC is higher than that absorbed by non-optimal controller; this is the price which has to be paid by the optimal controller to have better dynamic and tracking performance. Figure 14. Effects of control signal u based on optimal and non-optimal ASC controlled PAM system. Figure 15 combines the responses based on optimal ASC and optimal CSC. It is clear from the figure that the optimal ASC has less error and hence better tracking performance than optimal CSC. Tables 5 and 6 reports the numerical values for RMS values of errors and input pressures due to including and excluding the PSO technique. Based on Table 5, one can conclude that the RMSE value in the case of optimal ASC (0.0483 rad) is considerably smaller than that obtained by optimal CSC Figure 14. Effects of control signal u based on optimal and non-optimal ASC controlled PAM system. Figure 15 combines the responses based on optimal ASC and optimal CSC. It is clear from the figure that the optimal ASC has less error and hence better tracking performance than optimal CSC. Tables 5 and 6 reports the numerical values for RMS values of errors and input pressures due to including and excluding the PSO technique. Based on Table 5, one can conclude that the RMSE value in the case of optimal ASC (0.0483 rad) is considerably smaller than that obtained by optimal CSC (0.1686 rad). This indicates the dynamic and tracking performance of optimal ASC outperforms that resulting from optimal CSC. However, this superiority in performance of optimal ASC will be at the price of higher input pressure generated by optimal ASC (7.5494 × 10 6 Pa) as opposed to lower control signal (input pressure) produced by optimal CSC (1.1367 × 10 5 Pa).  for bicep muscle, respectively. The boundness of the estimation errors of uncertain parameters is one of the essential issues in evaluating the performance of adaptive design for most adaptive controllers. If the adaptive control lacks the ability to confine the estimation errors within certain bounds, then the stability of controlled system may be violated. Based on the figures, one can determine the bounds of to be ≤ 0.8 and to be ≤ 5.5 for 0 ≤ ≤ 60 s. One can conclude that ASC could successfully confine these coefficients within bounded values over the simulation time, and hence the adaptive controller could avoid the instability problem which may arise due to the drift of uncertainty in system parameters.   Figures 16 and 17 show the actual and estimated values of viscosity coefficient B 0b and spring coefficient K for bicep muscle, respectively. The boundness of the estimation errors of uncertain parameters is one of the essential issues in evaluating the performance of adaptive design for most adaptive controllers. If the adaptive control lacks the ability to confine the estimation errors within certain bounds, then the stability of controlled system may be violated. Based on the figures, one can determine the bounds of B 0b to be B 0b ≤ 0.8 and K 0 to be K 0 ≤ 5.5 for 0 ≤ t ≤ 60 s. One can conclude that ASC could successfully confine these coefficients within bounded values over the simulation time, and hence the adaptive controller could avoid the instability problem which may arise due to the drift of uncertainty in system parameters. adaptive controllers. If the adaptive control lacks the ability to confine the estimation errors within certain bounds, then the stability of controlled system may be violated. Based on the figures, one can determine the bounds of to be ≤ 0.8 and to be ≤ 5.5 for 0 ≤ ≤ 60 s. One can conclude that ASC could successfully confine these coefficients within bounded values over the simulation time, and hence the adaptive controller could avoid the instability problem which may arise due to the drift of uncertainty in system parameters.

Scenario III: Validation of Proposed Controller
Before proceeding in validation, it is worthwhile to note that the structure of the PAM-actuated single-link robot arm differs from its counterparts with the same link and actuation. In the present work, the cable attached to the pair of muscles that works to rotate the pulley is connected at the end of the robot arm, as shown in Figure 1. Therefore, in order to conduct a fair performance comparison, one has to direct the comparison study with the same robot structure controlled by an adaptive controller. As such, reference [16] has been chosen. However, one problem of the compared work is that it's proposed adaptive controller has been verified based on a different reference trajectory and different robot system parameters to those used in the present work. Therefore, to be consistent with the compared work, the following desired trajectory is adopted in this scenario: 50 kg , = 0.025 m . By this setting, the arm is allowed to travel between = 0° (arm fully straightened) to = 180° (arm fully bent). The initial joint angle is set to 42° (0.733 rad) for the controlled system based on both controllers. Figure 18 shows the tracking performance of the controlled PAM-actuated robot arm based on the optimal adaptive synergetic controller and the compared controller. Starting from the same initial

Scenario III: Validation of Proposed Controller
Before proceeding in validation, it is worthwhile to note that the structure of the PAM-actuated single-link robot arm differs from its counterparts with the same link and actuation. In the present work, the cable attached to the pair of muscles that works to rotate the pulley is connected at the end of the robot arm, as shown in Figure 1. Therefore, in order to conduct a fair performance comparison, one has to direct the comparison study with the same robot structure controlled by an adaptive controller. As such, reference [16] has been chosen. However, one problem of the compared work is that it's proposed adaptive controller has been verified based on a different reference trajectory and different robot system parameters to those used in the present work. Therefore, to be consistent with the compared work, the following desired trajectory is adopted in this scenario: θ d (t) = 60 • + 62.5 • (sin2π f 1 t + 0.05sin2π f 2 t) π 180 where f 1 = 0.01 Hz and f 2 = 0.1 Hz. In addition, the following parameters are used for the PAM-actuated robot arm based on proposed and compared controllers: L = 0.5 m, r = 0.05/π m, M = 50 kg, a = 0.025 m. By this setting, the arm is allowed to travel between θ = 0 • (arm fully straightened) to θ = 180 • (arm fully bent). The initial joint angle is set to 42 • (0.733 rad) for the controlled system based on both controllers. Figure 18 shows the tracking performance of the controlled PAM-actuated robot arm based on the optimal adaptive synergetic controller and the compared controller. Starting from the same initial joint condition, it is clear that the joint angular response based on optimal ASC is faster than that based on its counterpart. The angular position resulting from optimal ASC reaches the desired trajectory in 0.65 s, while it coincides with the desired trajectory in 2.5 s in the case of the compared controller. The calculation of RMSE is the other index of evaluation, which is used for comparison. It has been found that the RMSE due to optimal ASC is equal to 0.0147 rad, while this value is equal to 0.0278 rad in the case of the compared controller. This numeric report indicates that the proposed adaptive ASC gives better tracking performance than that given in the previous work.

Conclusions
This paper presents a new classical and adaptive control design based on a synergetic theory for tracking control of a single-link robot arm actuated by a pair of artificial muscles. The adaptive synergetic controller has been designed to cope with inherited uncertainties in muscle parameters. The design of the proposed controllers has been developed to guarantee the stability of the controlled robotic system.
According to the simulated results, one can conclude that the optimal CSC gives better tracking performance in terms of transient and steady-state characteristics than that obtained by non-optimal CSC. However, the energy drawn by the controlled system based on optimal CSC is higher than that based on the non-optimal controller. In the presence of uncertainty in muscle parameters, the tracking control performance of the adaptive controlled system based on optimal ASC is better than that based on non-optimal ASC. This will be at the expense of higher required input pressure to actuate the muscles in the case of optimal ASC. Moreover, the optimal ASC controller gives better tracking control performance than optimal CSC. Again, this superiority in performance obtained by optimal ASC was on account of higher absorbed energy needed to supply the actuating muscles. Moreover, the optimal ASC could successfully confine the estimation errors of uncertain parameters within certain bounds, which, in turn, will avoid the instability problem due to drifting in estimation errors. Another conclusion that can be drawn is that the proposed adaptive controller generates continuous and monotonic control signals. The proposed controller has been validated with previous work and it has been shown that the proposed controller gives better tracking performance in terms of tracking speed and error than the previous controller.
The future work of the work will focus on real-time implementation and verification of the proposed controller on an actual PAM-actuated robot arm. In addition, a comparison study can be conducted with another adaptive control scheme for the same robot structure, or the PSO technique can be used as a comparison format with other recent optimization techniques.

Conclusions
This paper presents a new classical and adaptive control design based on a synergetic theory for tracking control of a single-link robot arm actuated by a pair of artificial muscles. The adaptive synergetic controller has been designed to cope with inherited uncertainties in muscle parameters. The design of the proposed controllers has been developed to guarantee the stability of the controlled robotic system.
According to the simulated results, one can conclude that the optimal CSC gives better tracking performance in terms of transient and steady-state characteristics than that obtained by non-optimal CSC. However, the energy drawn by the controlled system based on optimal CSC is higher than that based on the non-optimal controller. In the presence of uncertainty in muscle parameters, the tracking control performance of the adaptive controlled system based on optimal ASC is better than that based on non-optimal ASC. This will be at the expense of higher required input pressure to actuate the muscles in the case of optimal ASC. Moreover, the optimal ASC controller gives better tracking control performance than optimal CSC. Again, this superiority in performance obtained by optimal ASC was on account of higher absorbed energy needed to supply the actuating muscles. Moreover, the optimal ASC could successfully confine the estimation errors of uncertain parameters within certain bounds, which, in turn, will avoid the instability problem due to drifting in estimation errors. Another conclusion that can be drawn is that the proposed adaptive controller generates continuous and monotonic control signals. The proposed controller has been validated with previous work and it has been shown that the proposed controller gives better tracking performance in terms of tracking speed and error than the previous controller.
The future work of the work will focus on real-time implementation and verification of the proposed controller on an actual PAM-actuated robot arm. In addition, a comparison study can be conducted with another adaptive control scheme for the same robot structure, or the PSO technique can be used as a comparison format with other recent optimization techniques.