Acta Polytechnica https://doi.org/10.14311/AP.2024.64.0519 Acta Polytechnica 64(6):519–529, 2024 © 2024 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague PERFORMANCE IMPROVEMENT OF MANIPULATOR ACTUATED BY PNEUMATIC ARTIFICIAL MUSCLES BASED ON SYNERGETIC CONTROL AND SOCIAL SPIDER OPTIMISATION ALGORITHM Alaq F. Hasan, Hadeer A. Raheem∗, Rajaa Hussein Middle Technical University, Technical Engineering Collage Baghdad, Mechatronics Engineering, 8998+QHJ, Baghdad, Iraq ∗ corresponding author: hadeerasaadtec@mtu.edu.iq Abstract. The manipulator actuated by pneumatic artificial muscles (PAM) is a widely used type of robotic arm in industrial automation. However, its performance can be limited by non-linear dynamics and uncertainties in the system. To overcome these limitations, this paper proposes a synergetic control strategy (SACT) to improve the performance of the SACT, a social spider optimisation algorithm (SSO) has been suggested for adjusting its parameters. To verify the performance of a PAM-actuated manipulator based on an optimal SACT controller, a computer simulation study was conducted using MATLAB software. Moreover, a comparison study between the optimal synergetic algorithm control theory and the optimal sliding mode controller (SMC) has been made in terms of robustness and transient behaviour characteristics. The provided simulation results have shown that the SACT controller exhibited quicker convergence towards the desired trajectory and maintained a lower steady- state error as compared to the SMC controller. Additionally, the SACT controller demonstrated more resilience to variations in parameters and showed more robust characteristics. Keywords: PAM-actuated manipulator, synergetic algorithm control theory, sliding mode controller, social spider optimisation algorithm. 1. Introduction The actuator is an essential part of some systems that is responsible for transferring energy from one form to another, e.g. the mechanical motion interprets an explicit nature of any interaction between the parts of a system and its environment. A recent study in robotic systems technology suggests that the interface format, or mechanism movement, could be changed from geared motors to a softer biological idea of bone- tendon-muscle. This trend will ensure that machines and humans interact closely, as well as that there is safety, redundancy, self-repair, functional and physical softness. As a result, any modern actuation system must include the advantageous characteristics of tra- ditional actuators, including correct design features, such as the ratio between the force and the actuator volume and extra operational life. The pneumatic actuators were one of these modern actuation tech- nologies, such as cylinders, which are widely used in factory automation systems. Recently, robotics has begun to use pneumatics as the main power source of motion. One of the main advantages of pneumatic actuators is their low weight and their inherent ac- commodating behaviour as compared to hydraulic actuators [1–3]. One of the lesser known types of pneumatic ac- tuator is the so-called pneumatic artificial muscle. These are the inverse form of the bellows, i.e. they have some similarities in the inflation process. Their force is dependent on pressure, in addition to their dependence on the state of inflation, and that cre- ates a second source of spring-like behaviour. PAM’s weight is extremely light because the element of their core is a membrane, thus, compared to cylinders, they can transfer the same amount of energy, as long as they are operating at the same ranges of pressure and volumes. For these reasons, PAMs have great poten- tial and advantages to be used in bio-robotic systems with reliable performance [4]. Pneumatic muscle-based systems are characterised by high complexity, nonlinearity, and uncertainty in their parameters. Several researchers have suggested various control schemes to tackle the control challenges of pneumatic muscle-based systems. The following studies address the control techniques commonly used to control PAM-actuated systems: • Al-Jodah and Khames [5] presented two control strategies to control the angular position of a single link robotic arm actuated by a pair of PAMs. The first one is sliding mode control based on a super- twisting algorithm and the other is a classic sliding mode control. The main objective of this work is to reduce the effect of the chattering problem that is commonly associated with SMC design. To solve this problem, the researcher used the super twisting- algorithm. The computer simulation has verified a low tracking error in terms of a sliding mode control based on the super-twisting algorithm as compared to classical sliding mode control. More- over, a sliding mode control based on the super- 519 https://doi.org/10.14311/AP.2024.64.0519 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en A. F. Hasan, H. A. Raheem, R. Hussein Acta Polytechnica twisting algorithm shows a solid durability against uncertainty in system parameters. • Scaff et al. [6] used a PID controller to control the angular position of a single link robotic arm ac- tuated by McKibben PAMs and to improve the performance of the PID controller, they used Simu- lated Optimisation Algorithm (SOA). This study did not show the efficiency of the controlled system in the presence of variation in the parameters of the system. • Boudoua et al. [7] presented a design of a super- twisting sliding mode controller based on neural network to control the angular position of a PAM robotic arm. To estimate the unknown dynamics of the robot, the study used a two-neural network layer with online adaptive learning law. The work is lacking in the capability to remove the chattering problem in the control signal. • Repperger et al. [8] applied the gain-scheduled con- troller to control the angular position of a large-scale pneumatic muscle actuator. Through numerous ex- periments of transient and steady-state responses, the scheduled gains had been established. The study did not show the efficiency of the proposed controller in the presence of variations in the parameters of the system. • Enzevaee et al. [9] proposed a fuzzy logic controller based on an Active Force Control technique to con- trol the angular position of a manipulator with one degree of freedom. The obtained results of the proposed controller show the controller’s ability to compensate for the subjected disturbances robustly. • Farag and Azlan [10] applied the adaptive backstep- ping controller to control the angular position of Anthropomorphic Robot Hand at PAM actuators. The adaptive controller has been integrated with the PID controller to form a hybrid controller. The adaptive law has been obtained to estimate the un- certainty in the values of the system parameters. The simulation results have validated the tracking performance based on the proposed controller. The synergetic control strategy theory is based on a state-space theory that focuses on the design and control of extremely complicated systems. This con- trol strategy can allow the system’s states to develop on the designer’s selected invariant manifolds while achieving the intended performance in the presence of uncertainties and disruptions [11]. In view of this point, a brief discussion of a research relevant to the synergetic control strategy theory will be presented: • Nechadi et al. [12] proposed a synergetic control strategy to satisfy the output voltage of the DC-DC Boost converter at the required level. The proposed controller shows high performance competence de- spite the nonlinear behaviour of the system. • Al-Dujaili et al. [13] investigated an adaptive control design using adaptive synergetic control strategy Figure 1. One degree of freedom manipulator actu- ated by PAMs. theory to mitigate the effect of vibrations in a build- ing structure due to earthquake action. The study involved a single-degree-of-freedom building, and numerical simulation demonstrated the ability of the proposed scheme to significantly reduce vibra- tions from the impact of the earthquake • Al-Khazraji et al. [14] presented the design of syner- getic control strategy theory to improve the position tracking accuracy of a ball and beam system. Com- pared to a classic state feedback controller, the synergetic control strategy theory exhibited better performance in terms of stability, robustness char- acteristics, and finite-time convergence of position error. The major contributions addressed by this work can be summed up as: • developing a SACT to control the angular position of a PAM-actuated single link robotic arm, • optimal tuning of SACT parameters to improve the performance of an SACT-based PAM-actuated manipulator, • carrying out a comparative study between SACT and SMC algorithms in terms of robustness and transient behaviour characteristics. Subsequent sections of this paper are organised as follows: Section 2 presents the modelling of a PAM- actuated manipulator. Section 3 presents the de- sign and stability analysis of SACT and SMC algo- rithms. In addition, this section presents the optimi- sation problem and its SSO-based solution. Section 4 presents a computer simulation to validate the per- formance of the proposed controllers. Finally, the conclusions reached based on the results of the numer- ical simulation has been highlighted in Section 5. 2. Mathematical model The mathematical model of a PAM-actuated single link robotic arm had been derived and performed previously in [5, 15]. Figure 1 illustrates the general 520 vol. 64 no. 6/2024 Performance improvement of manipulator actuated by pneumatic . . . concept of the PAM-actuated single link robotic arm whereas its equation of motion is as follows: (ml2 + i)φ̈ + mgl cos φ = T, (1) where the mass at the end point of the robotic arm is denoted by symbol m, the length of the arm is represented by symbol l, the gravitational acceleration is denoted by symbol g, the rotation angle and angular acceleration of the arm are denoted by the symbols φ and φ̈, respectively, the moment of inertia for the robotic arm is denoted by symbol i, and lastly, T , which stands for the generated torque by the bicep and tricep PAMs and required to rotate the link. The torque equation is given as: T = (Ft(.) − Fb(.))r, (2) Where, r is the radius of the pulley and Ft(.), Fb(.) are the tricep and bicep forces that are generated from PAMs, respectively. The mathematical model of PAMs [5], which have been used to find Ft(.) and Fb(.), had been derived as follows: Ft(.) = −Kt(xt)xt − Bt(ẋt)ẋt + Pt, (3) Fb(.) = −Kb(xb)xb − Bb(ẋb)xb + Pb, (4) where (t, b) stand for the tricep and bicep PAMs, respectively. Hence, Kt(xt) and Kb(xb) are the non- linear function coefficients of the spring of the tricep and bicep PAM positions, respectively. Bt(ẋt) and Bt(ẋt) are the nonlinear function coefficients of the damper of the tricep and bicep PAM velocities, respec- tively. xt and xb are the amount of contraction in the bicep and tricep PAMs, respectively. ẋt and ẋb are the velocities of tricep and bicep PAMs in the contraction case, respectively. Pt and Pb are the pressures for the two PAMs [4]. By taking into consideration i = (t, b), then Ki(xi) and Bi(ẋi) are given by: Ki(xi) = k1ixi 2 + k2ixi + k3i, (5) Bi(ẋi) = b1iẋ 2 i + b2iẋi + b3i, (6) where k1i, k2i, k3i, b1i, b2i, b3i are constants. There are two states of work for these constants, the first is when the PAM is inflated, and the second is when the PAM is in the deflated state. The state of the bicep and tricep PAMs can be explained as follows: φ̇ < 0 in case of { bicep inflacted, tricep deflated, φ̇ > 0 in case of { bicep deflacted, tricep inflated. (7) Thus, the bicep and tricep PAM pressure equations can be written as: Pt = P0t + ∆P, (8) Pb = P0b + ∆P, (9) where P0t and P0b are considered to be the initial pres- sures. ∆P is the difference between the two pressures of the tricep and bicep which would be the system’s control input. Now, xt and xb are the amounts of muscle contraction calculated as: xt = (π 2 + φ)r, ẋt = rφ̇, (10) xb = (π 2 − φ)r, ẋb = −rφ̇, (11) where π 2 is presumed to be the null position at which the value of xt and xb is zero. Now, substitute Equa- tions (8) and (9) in Equations (3) and (4), then sub- stitute the resulting two equations in Equation (2), and the resulting equation is as follows: T = ( − Kt(xt)xt − Bt(ẋt)ẋt + P0t + ∆P + Kb(xb)xb + Bb(xb)xb − P0b + ∆P ) r. (12) Then, make the substitution of both Equations (5) and (6) in Equation (12): T = ( − (k1tx 2 t + k2txt + k3t)xt − (b1tẋ 2 t + b2tẋt + b3t)ẋt + P0t + ∆P + (k1bx2 b + k2bxb + k3b)xb + (b1bẋ2 b + b2bẋb + b3b)ẋb − P0b + ∆P ) r. (13) Finally, substitute Equations (10) and (11) in Equa- tion (13), then substitute the resulting equation in Equation (1), and the resulting equation is as follows: φ̈ = δ1φ3 + δ2φ2 + δ3φ + δ4φ̇3 + δ5φ̇2 + δ6φ̇ + δ7 cos φ + δ8 + bu, u = ∆P, δ1 = −r3(k1b + k1t) ml2 + i , δ2 = 3πr4 2 (k1b − k1t) + r3(k2b − k2t) ml2 + i , δ3 = − 3πr4 2 (k1b − k1t) ml2 + i + πr3 (k2b + k2t) + r2(k3b + k3t) ml2 + i , δ4 = −r4(b1b + b1t) ml2 + i , δ5 = r3(b2b − b2t) ml2 + i , δ6 = −r2(b3b − b3t) ml2 + i , δ7 = glm ml2 + i , δ8 = r(P0b − P0t) + πr4 8 (k1t − k1b) ml2 + i + πr3 4 (k2t − k2b) + πr2 2 (k3t − k3b) ml2 + i , b = 2r ml2 + i .  (14) 521 A. F. Hasan, H. A. Raheem, R. Hussein Acta Polytechnica The system can be represented as follows, taking the uncertainty into consideration: φ̈ = f0 + ∆f + bu, f0 = δ1φ3 + δ2φ2 + δ3φ + δ4φ̇3 + δ5φ̇2 + δ6φ̇ + δ7 cos φ + δ8, ∆f = ∆δ1φ3 + ∆δ2φ2 + ∆δ3φ + ∆δ4φ̇3 + ∆δ5φ̇2 + ∆δ6φ̇ + ∆δ7 cos φ + ∆δ8,  (15) where δi is the nominal values and ∆δi represent the uncertainty in the value of the system parameters. 3. Control theory Two control theories have been used in this work to control the system with a modern optimisation algorithm to obtain the optimal parameter for the controllers. 3.1. Sliding mode control (SMC) algorithm In sliding mode control, the control signal can typically be broken down into two parts. The first part is known as “the equivalent control”, which represents both the sliding surface and the system dynamics. The other part is referred to as “the switching control”, which maintains the system dynamics on the sliding surface [16, 17]. Firstly, the sliding surface could be defined as: s = ce + ė, (16) where s and c are the sliding surface and the sliding coefficient, respectively, whereas e and ė can be defined as the tracking error and the tracking error derivative, respectively. These two parameters are formulated as:{ e = φ − φr, ė = φ̇ − φ̇r, (17) where φ and φr are the actual trajectory and the desired trajectory, respectively. The time derivative of Equation (16) is: ṡ = ë + cė, ṡ = φ̈ − φ̈r + cė. (18) As previously mentioned, the control law consists of two parts, equivalent and switching, thus: u = 1 b (ueq + usw) . (19) Equivalent control, ueq, was proposed by Filipov as a means of controlling the system without taking external disturbances and uncertainties into account. By setting the sliding surface’s derivative equation to zero, to keep the controlled variable on the surface, the equivalent component can be determined based on the equation: ueq = −fo + φ̈r − cė. (20) In conventional SMCs, switching control usw is usu- ally defined as: usw = −k sign(s), (21) where k and k sign(s) are the positive constant and symbolic function, respectively, that is defined by: sign(s) =  −1 if s < 0, 0 if s = 0, +1 if s > 0. (22) We can perform the control law as: u = 1 b (−fo + φ̈r − cė − k sign(s)). (23) For emphasising the asymptotic stability of the PAM system via SMC, the Lyapunov function has been determined as: V = 1 2s2. (24) Taking the first derivative of time for Equation (24), we get: V̇ = sṡ, V̇ = s (fo + ∆f + bu − q̈r + γė) . (25) Assumption 1: Parametric uncertainties in the sys- tem ∆f are assumed to be unknown but bounded |∆f | ≤ fmax. Now, substituting Equation (23) into Equation (25), we get: V̇ = s (−k sign(s) + ∆f) . (26) Remark 1: According to Equation (26), the stabil- ity of the PAM system based on SMC is ensured if k is chosen such that k ≥ fmax ≥ |∆f |. To mitigate the problem of chattering, which is a common issue in a conventional sliding mode con- trol, an approximation signum function is used as a substitute for the sign function [18]: sign(s) ≈ 2 π tanh(βs), (27) where β is a controller design parameter. Now, we can perform the control law as: u = 1 b ( −fo + φ̈r − cė − 2k π tanh(βs) ) . (28) Figure 2 shows the control scheme of a PAM- actuated manipulator system based on the sliding mode controller. 3.2. Synergetic control approach Kolesnikov [19] introduced the synergetic control ap- proach, which involves starting the synergetic synthe- sis process by selecting a macro variable. This variable is a function of the system states: σ = σ(x, t). (29) 522 vol. 64 no. 6/2024 Performance improvement of manipulator actuated by pneumatic . . . Figure 2. PAM-actuated manipulator controlled by SMC. The macro-variable value is denoted by σ and this variable can be determined by a user-defined function, σ(x, t), a function of time and system state variable. Thus, the designer of the control has the flexibility of selecting the characteristics of the macro-variable based on various factors, such as response time, control parameters, limitations of the command, etc. The main objective of the synergetic control approach is to push the system states to work on the σ = 0 manifold [12, 20, 21]. That helps the desired macro- variable dynamic evolution to be determined as: λσ̇ + σ = 0. (30) When the macro-variable becomes equal to zero, design parameter λ determines the speed at which the closed-loop system converges to the manifold. But, σ̇ defines the derivative of the aggregated macro- variable. Then, the solution of the differential equation is obtained as the following function: σ(t) = σ0 e− t λ . (31) σ(t) is attracted to σ = 0 from any initial condition σ0. The chain rule of differentiation gives: σ̇ = dσ dx dx dt . (32) By combining Equations (30) and (32), we obtain: λ dσ dx f(x, u, t) + σ = 0. (33) After solving Equation (33) for u, the controller law can be expressed as: u = g(x, σ, λ, t). (34) The equation demonstrates that the controller’s characteristics can be selected by the designer through the choice of a particular control parameter λ and a proper macro-variable. For controlling a PAM system by using a synergetic control algorithm, the following macro-variable has been considered: σ = ė + γe. (35) Now, by using the substitution of Equation (35) into Equation (30), we obtain: λ(ë + γė) + σ = 0, (36) then, by substituting Equation (15) into Equation (36) with rearranging, we obtain: λ(q̈ − q̈r + γė) + σ = 0, λ(fo + ∆f + bu − q̈r + γė) + σ = 0. Thus, the control action equation becomes: u = 1 b (−fo + q̈r − γė) − σ λb . (37) For emphasising the asymptotic stability of the PAM system via SACT, the below Lyapunov function has been selected: V = 1 2σ2. (38) By taking the derivative of time for Equation (38), we get: V̇ = σσ̇, V̇ = σ (fo + ∆f + bu − q̈r + γė) , (39) and by substituting Equation (37) into Equation (39), we get: V̇ = σ ( −σ λ + ∆f ) . (40) Note that for all t ≥ 0, V is positive definite and V̇ is negative definite if and only if σ λ ≥ fmax ≥ |∆f |. Figure 3 shows the control scheme of a PAM-actuated manipulator system based on the synergetic control approach. 3.3. Parameters optimisation based on social spider optimisation algorithm The SACT design parameters must be modified to provide the optimum controller performance for the PAM-actuated manipulator system. The trial-and- error approach of determining or modifying these pa- rameters is inefficient and fails to produce optimal 523 A. F. Hasan, H. A. Raheem, R. Hussein Acta Polytechnica Figure 3. PAM-actuated manipulator controlled by SACT. results in the form of more effectively dynamic perfor- mance of controlled systems. As a result, the Social Spider Optimisation (SSO) technique has been pro- posed as a method to discover the ideal values of these parameters in order to ensure excellent performance of the proposed controllers in terms of dynamic response. The design parameters for SMC are (c, k, β), while for SACT, they are (γ, λ). Erik Cuevas et al. [22] established the SSO method, which models the cooperative behaviour of social spi- ders. SSO considers two search agents (spiders), one male and one female. Each individual is treated with a unique set of evolutionary operators based on its gen- der, which simulates various cooperative behaviours present in the colony. The flowchart of the SSO algo- rithm is shown in Figure 4. Like many metaheuristic algorithms, the performance of SSO depends on its parameter settings, such as population size, neigh- bourhood size, and movement parameters. These parameters often require fine-tuning to achieve good performance on different optimisation problems. In our design, the Integral of Time-Weighted Absolute Error (ITAE) was used to evaluate the cost of each in- dividual during the search for choosing the best value with minimum cost for all controllers. The mathemat- ical formula of ITAE is shown in Equation (41). The SSO algorithm has been run by setting several itera- tions equal to 32, and 60 for the number of spiders. f = ITAE = ∫ T 0 t · |e| dt. (41) 4. Computer simulation The physical parameters of the single link robotic arm actuated by PAMs are listed in Table 1 [5, 15]. A model of the robot incorporating the proposed con- trollers has been created using MATLAB/Simulink and the “Ode45” is used as the numerical solver. The results of the optimisation process based on the SSO are given in Table 2. Also, the behaviour of the cost function during the entire optimisation process with the design parameters for both the SACT and SMC controllers is shown in Figure 5. The desired input signal applied for the controlled system is given by equation: φr = π 2 + 0.5 (sin(2πf1t) + sin(2πf2t) + sin(2πf3t)) , (42) where, f1 = 20 mHz, f2 = 50 mHz and f3 = 90 mHz. The response of the robotic system with track- ing behaviour for an angular position in terms of an uncertainty-free system is shown in Figure 6 and the angular velocity response of the same descriptive system is illustrated in Figure 7. Figures 8 and 9 show the behaviour of the control signals and the error sig- nals resulting from the used controllers, respectively. The control system’s performance based on SACT and SMC is reported in Table 3. The evaluation of the performance is based on root mean square error, steady-state errorm and settling time. From the pre- vious results, we can see that the PAM system based on SACT was able to complete the trajectory tracking with high motion accuracy as compared to the one based on SMC. After evaluating the results in terms of an uncertainty-free system, an uncertainty within 10 % of the nominal value was applied in the second case. Fig- ures 10 and 11 show the tracking performance of both the angular position and velocity through the two controllers used sequentially. While Figures 12 and 13 show the behaviour of the control signal and error signal that have resulted from the applied controllers. Hence, from the provided results, it can clearly be seen that the SMC controller has failed to maintain the stability of the system as compared to the SACT controller. 5. Conclusion In this work, a synergetic algorithm control theory and a sliding mode control method are developed for tracking PAM system control. These two control the- ories with their control laws were used to maintain the asymptotic stability of the system. The robust- ness and effectiveness of the proposed control schemes have been verified by several simulation studies. Ac- cording to the simulation results, the SACT showed 524 vol. 64 no. 6/2024 Performance improvement of manipulator actuated by pneumatic . . . Figure 4. SSO flowchart. Parameter Value Unit m 20 kg l 0.5 m I 1.667 kg m−2 g 9.81 m s−2 r 0.05 m P0b 400 kPa P0t 400 kPa Table 1. The parameters of single link robotic arm. Controller Optimal values Coefficient Value SACT γ 1.4558 λ 0.0055 SMC c 1.5278 k 0.9997 β 17.6440 Table 2. The proposed values of the design parame- ters for SACT and SMC by the SSO algorithm. 525 A. F. Hasan, H. A. Raheem, R. Hussein Acta Polytechnica (a). SMC. (b). SACT. Figure 5. Evolution of the cost function versus the iterations. Controller RMSE Settling time [sec] Steady-state error [rad] ITAE SACT 0.0499 3.1 1.5 × 10−5 324.8 SMC 0.1080 4.6 9.8 × 10−5 1 732 Table 3. Evaluation of PAM-actuated manipulator controlled using SACT and SMC. Figure 6. The behaviour of angular position without uncertainty. Figure 7. The behaviour of angular velocity without uncertainty. Figure 8. The behaviour of control action without uncertainty. Figure 9. The behaviour of angle error signal without uncertainty. 526 vol. 64 no. 6/2024 Performance improvement of manipulator actuated by pneumatic . . . Figure 10. The behaviour of angle error signal with 10 % uncertainty in parameters. Figure 11. The behaviour of angular position with 10 % uncertainty in parameters. Figure 12. The behaviour of angular velocity with 10 % uncertainty in parameters. Figure 13. The behaviour of control action with 10 % uncertainty in parameters. a better tracking performance and super robust char- acteristics against variations of the parameter’s values than the SMC. The sample results obtained show that the SACT has a lower root mean square error (RMSE), 53.79 %, than the SMC. In addition, a fur- ther performance improvement was achieved with the two presented control schemes by using the SSO algo- rithm. Future research endeavours of this study; one can suggest a two or three degree of freedom robot manip- ulator and use the same control techniques to show the robustness of SACT. Another suggestion is to use different optimisation techniques, such as the Whale Optimisation Algorithm (WOA) [23], Spotted Hyena Optimiser (SHO) [24], and Sparrow Search Optimi- sation [25] to make a comparison with those used in this study. One can propose other control strategies, such as Model Reference Adaptive Control (MRAC), observer-based control, robust control, backstepping control, Active Disturbance Reject Control (ADRC), and nonlinear PD control to further improve the pro- posed controller as future extension of this study [26– 37]. References [1] R. M. Robinson, C. S. Kothera, R. M. Sanner, N. M. Wereley. Nonlinear control of robotic manipulators driven by pneumatic artificial muscles. IEEE/ASME Transactions on Mechatronics 21(1):55–68, 2016. https://doi.org/10.1109/TMECH.2015.2483520 [2] H. Khajehsaeid, B. Esmaeili, R. Soleymani, A. Delkhosh. Adaptive back stepping fast terminal sliding mode control of robot manipulators actuated by pneumatic artificial muscles: continuum modelling, dynamic formulation and controller design. Meccanica 54(8):1203–1217, 2019. https://doi.org/10.1007/s11012-019-01012-4 [3] M. Khaleel Ahmed, M. Majid Msallam. 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IIUM Engineering Journal 25(2):367–380, 2024. https://doi.org/10.31436/iiumej.v25i2.3188 529 https://doi.org/10.1080/01969722.2020.1758467 https://doi.org/10.1109/ICPCSI.2017.8392309 https://doi.org/10.1155/2019/7689673 https://doi.org/10.3390/app10124069 https://doi.org/10.17509/ijost.v6i2.36275 https://doi.org/10.31436/iiumej.v25i2.3188 Acta Polytechnica 64(6):519–529, 2024 1 Introduction 2 Mathematical model 3 Control theory 3.1 Sliding mode control (SMC) algorithm 3.2 Synergetic control approach 3.3 Parameters optimisation based on social spider optimisation algorithm 4 Computer simulation 5 Conclusion References