3.1. PIDSMC design

PIDSMC is a PID sliding surface SMC method that is used to select the sliding surface in order to increase the robustness and sensitivity of the SMC. The surface of PIDSMC is as follows:

(6) \begin{equation}\sigma \left(t\right)=K_{p}e\left(t\right)+K_{i}\int e(t)dt+K_{d}\frac{d}{dt}e(t)\end{equation}

where $K_{p}$ , $K_{i}$ and $K_{d}$ are the $6\times 6$ diagonal gain matrix for terms of proportional, integral, and derivative, respectively. The trajectory tracking position error is:

(7) \begin{equation}e\left(t\right)=q_{d}\left(t\right)-q_{a}(t)\end{equation}

where $q_{d}(t)$ and $q_{a}(t)$ represent the desired and actual angular positions of the manipulator’s end-points, respectively.

Typically, the design of the PIDSMC sliding surface is predicated upon the position error ( $e)$ ) and its velocity ( $\dot{q}$ ). The convergence of the sliding surface derivative, $\dot{\sigma }(t)\rightarrow 0$ , is a critical requirement for achieving stability in the sliding mode. This convergence, in turn, guarantees that the tracking error also approaches zero within the sliding mode. This means the system’s actual trajectory aligns with the desired trajectory, and only a steady-state control effort is required to maintain this alignment. The sliding surface $\sigma (t)$ is differentiated with respect to time from the equation:

(8) \begin{equation}\dot{\sigma }\left(t\right)=K_{p}\dot{e}\left(t\right)+K_{i}e(t)+K_{d}\ddot{e}(t)\end{equation}

By replacing $\ddot{e}(t)=\ddot{q}_{d}(t)-\ddot{q}_{a}(t)$ with the equality in Eq. (8), the primary derivative of $\sigma (t)$ is ascertained as:

(9) \begin{equation}\dot{\sigma }\left(t\right)=K_{p}\dot{e}\left(t\right)+K_{i}e(t)+K_{d}\left(\ddot{q}_{d}\left(t\right)-\ddot{q}_{a}(t)\right)\end{equation}

Substituting the angular acceleration from Eq. (4) into Eq. (9) for $\ddot{q}_{a}(t)$ Eq. (9) becomes as follows:

(10) \begin{align}\dot{\sigma }\left(t\right)&=K_{p}\dot{e}\left(t\right)+K_{i}e\left(t\right)+K_{d}\left(\ddot{q}_{d}\left(t\right)-M^{-1}\left(q\right)u\left(t\right)+M^{-1}\left(q\right) C\left(q,\dot{q}\right)\dot{q}+M^{-1}\left(q\right)G\left(q\right)\right.\nonumber\\& \quad\left. +M^{-1}\left(q\right)\tau _{d}\left(q,\dot{q}\right)\right)\end{align}

$\dot{\sigma }(t)$ is adjusted to zero to obtain the equivalent control input $u_{eq}(t)$ given by the equation below:

(11) \begin{align}u_{eq}\left(t\right)&=\left(M^{-1}\left(q\right)\right)^{-1}\left(\ddot{q}_{d}\left(t\right)+M^{-1}\left(q\right) C\left(q,\dot{q}\right)\dot{q}+M^{-1}\left(q\right)G\left(q\right)+M^{-1}\left(q\right)\tau _{d}\left(q,\dot{q}\right)\right.\nonumber\\ &\quad\left. +\frac{K_{p}}{K_{d}}\dot{e}\left(t\right)+\frac{K_{i}}{K_{d}}e(t)\right)\end{align}

The reaching law, delineated in the preceding equation, constitutes a component of the SMC control principle. Its design is aimed at guaranteeing the stability of the closed-loop system. The output of the system is forced to follow the sliding surface with the help of the reaching law. There are commonly used reaching laws in the literature, such as exponential, fixed rate, and power rate rules [Reference Liu and Wang36]. Due to the structure of the SMC, chattering is an undesirable phenomenon that adversely affects the efficient operation of the controller. The control signal of the SMC becomes a discontinuous switching signal on the sliding surface, which produces chattering. To mitigate or eliminate chattering in SMC, a narrow boundary layer can be strategically designed around the switching surface. For this, saturation [Reference Liu, Liu, Meng and An37] or a more softened sigmoid function [Reference Ataşlar-Ayyıldız, Karahan and Yılmaz22] can be used instead of the sign function. In this research, the switching input $u_{sw}(t)$ is selected as a sigmoid function, as given below:

(12) \begin{equation}u_{sw}\left(t\right)=\omega \frac{{\sigma} \left(t\right)}{\left| {\sigma} \left(t\right)\right| +\gamma }\end{equation}

where $\gamma \gt 0$ is the boundary layer thickness and $\omega \gt 0$ is the coefficient that determines how fast the control signal arrives at the sliding surface along the reaching phase. The complete control rule of PIDSMC is the combination of the equivalent control $u_{eq}(t)$ and the reaching control $u_{sw}(t)$ , as given below:

(13) \begin{align}u_{\textit{PIDSMC}}\left(t\right)&=u_{eq}\left(t\right)+u_{sw}\left(t\right)=\left(M^{-1}\left(q\right)\right)^{-1}\left(\ddot{q}_{d}\left(t\right)+M^{-1}\left(q\right) C\left(q,\dot{q}\right)\dot{q}+M^{-1}\left(q\right)G\left(q\right)\right.\nonumber\\&\quad\left. +M^{-1}\left(q\right)\tau _{d}\left(q,\dot{q}\right)+\frac{K_{p}}{K_{d}}\dot{e}\left(t\right)+\frac{K_{i}}{K_{d}}e\left(t\right)+\omega \frac{{\sigma} \left(t\right)}{\left| {\sigma} \left(t\right)\right| +\gamma }\right)\end{align}

3.2. FOPIDSMC design

In contrast to conventional or intelligent controllers, this control technique draws upon fractional calculus. This mathematical framework offers a richer set of control design possibilities due to its inherent additional degrees of freedom. Leveraging fractional calculus, the proposed control strategy achieves enhanced closed-loop response and demonstrably robust closed-loop control performance, as corroborated by [Reference Ataşlar-Ayyıldız, Karahan and Yılmaz22].

Approaches such as Riemann-Liouville, Caputo, Grunwald-Letnikov, and Oustaloup are commonly used for fractional calculus analysis. Among fractional derivative definitions, the Riemann-Liouville approach enjoys the most widespread use. In the time domain, the integral and fractional derivative of $f(t)$ are given as [Reference Das38]:

(14) \begin{equation}{b}^{ }{D}{}_{t}^{\alpha }f\left(t\right)=\frac{1}{P\left(\varepsilon -\alpha \right)}\frac{d^{\varepsilon }}{dt^{\varepsilon }}\int _{b}^{t}\frac{f\left(\tau \right)}{\left(t-\tau \right)^{\alpha -\varepsilon +1}}d\tau\end{equation}

where $\varepsilon -1\lt \alpha \lt \varepsilon, \varepsilon \in \mathbb{N}, \alpha \in \mathbb{R}^{+}$ and the well-recognized Euler’s Gamma function is denoted by $P(\cdot )$ as follows:

(15) \begin{equation}P\left(x\right)=\int _{0}^{\mathrm{\infty }}t^{x-1}e^{-t}dt,\mathbb{R}\left(x\right)\gt 0\end{equation}

The fractional integral operator and the fractional derivative operator from the Riemann-Liouville definition are given by ${b}^{ }{D}{}_{t}^{-\lambda }f(t)$ and ${b}^{ }{D}{}_{t}^{\mu }f(t)$ , respectively, where $\lambda$ and $\mu$ positive real numbers. To simplify the expression, replace ${b}^{ }{D}{}_{t}^{-\lambda }f(t)$ and ${b}^{ }{D}{}_{t}^{\mu }f(t)$ with $D^{-\lambda }$ and $D^{\mu }$ .

Researchers have recently preferred to use fractional calculus to enhance the productivity and robustness of the PID controller, which is widely used in many systems. To do this, they have developed the FO PID (FOPID) controller, which is demonstrated as $PI^{\lambda }D^{\mu }$ , where $\lambda$ and $\mu$ correspond to the integral and derivative FOs, respectively. The differential form of the $PI^{\lambda }D^{\mu }$ controller is expressed as:

(16) \begin{equation}\sigma \left(t\right)=K_{p}e\left(t\right)+K_{i}D_{t}^{-\lambda }e\left(t\right)+K_{d}D_{t}^{\mu }e\left(t\right)\end{equation}

The FOPID structure given above is selected as the sliding surface in the SMC [Reference Kumar, Azar, Kumar and Rana30]. A vanishing second-order derivative of the sliding surface ( $\ddot{\sigma }(t)\rightarrow 0$ ) implies convergence of both error and its derivate (velocity) within the sliding mode. This additional constraint usually leads to faster reaching to the sliding mode compared to PIDSMC, resulting in quicker tracking of the desired trajectory. Differentiating Eq. (16) twice:

(17) \begin{equation}\ddot{\sigma }\left(t\right)=K_{p}\ddot{e}\left(t\right)+K_{i}D_{t}^{-\lambda +2}e\left(t\right)+K_{d}D_{t}^{\mu +2}e(t)\end{equation}

Substituting $\ddot{e}(t)=\ddot{q}_{d}(t)-\ddot{q}_{a}(t)$ into Eq. (17):

(18) \begin{equation}\ddot{\sigma }\left(t\right)=K_{p}\left(\ddot{q}_{d}\left(t\right)-\ddot{q}_{a}(t)\right)+K_{i}D_{t}^{-\lambda +2}e\left(t\right)+K_{d}D_{t}^{\mu +2}e(t)\end{equation}

Substituting the angular acceleration expressed by Eq. (4) for $\ddot{q}_{a}(t)$ in Eq. (18) yields the following:

(19) \begin{align}\ddot{\sigma }\left(t\right)&=K_{p}\left(\ddot{q}_{d}\left(t\right)-M^{-1}\left(q\right)\tau +M^{-1}\left(q\right) C\left(q,\dot{q}\right)\dot{q}+M^{-1}\left(q\right)G\left(q\right)+M^{-1}\left(q\right)\tau _{d}\left(q,\dot{q}\right)\right)\nonumber\\&\quad +K_{i}D_{t}^{-\lambda +2}e\left(t\right)+K_{d}D_{t}^{\mu +2}e(t)\end{align}

Setting $\ddot{\sigma }(t)=0$ to achieve the sliding mode condition allows us to obtain the equivalent control input, $u_{eq}(t)$ :

(20) \begin{align}u_{eq}\left(t\right)&=\left(M^{-1}\left(q\right)\right)^{-1}\left(\ddot{q}_{d}\left(t\right)+M^{-1}\left(q\right) C\left(q,\dot{q}\right)\dot{q}+M^{-1}\left(q\right)G\left(q\right)+M^{-1}\left(q\right)\tau _{d}\left(q,\dot{q}\right)\right.\nonumber\\&\quad\left. +\frac{K_{d}}{K_{p}}D_{t}^{\mu +2}e(t)+\frac{K_{i}}{K_{p}}D_{t}^{-\lambda +2}e\left(t\right)\right)\end{align}

The sigmoid function in Eq. (12) is selected as the switching law, as in the PIDSMC controller, and combined with Eq. (20) to obtain the total control signal for FOPIDSMC.

\begin{equation*} u_{\textit{FOPIDSMC}}\left(t\right)=u_{eq}\left(t\right)+u_{sw}\left(t\right) \end{equation*}

(21) \begin{align}&=\left(M^{-1}\left(q\right)\right)^{-1}\left(\ddot{q}_{d}\left(t\right)+M^{-1}\left(q\right) C\left(q,\dot{q}\right)\dot{q}+M^{-1}\left(q\right)G\left(q\right)+M^{-1}\left(q\right)\tau _{d}\left(q,\dot{q}\right)+\frac{K_{d}}{K_{p}}D_{t}^{\mu +2}e\left(t\right)\right.\nonumber\\&\quad\left. +\frac{K_{i}}{K_{p}}D_{t}^{-\lambda +2}e\left(t\right)+\omega \frac{{\sigma} \left(t\right)}{\left| {\sigma} \left(t\right)\right| +\gamma }\right)\end{align}

Thus, the FO PID surface effect is also included in the newly designed SMC-based controller.

3.3. FPIDSMC design

FL is particularly effective at overcoming the challenges of nonlinear system modeling and control and has therefore been widely adopted in control engineering applications. Just like conventional controllers, fuzzy controllers can be designed as Fuzzy PD, Fuzzy PI, Fuzzy PID, or a combination of these. FL integration with PID control has gained traction recently due to its demonstrated effectiveness in handling the intricacies of nonlinear and complex dynamic systems. Fuzzy PID controllers combine fuzzy PD and PI elements, with gains acting as scaling factors [Reference Woo, Chung and Lin39–Reference Yesil, Güzelkaya and Eksin40]. By choosing the fuzzy PID on the sliding surface of the SMC, the performance of the SMC increases with the adaptive coefficients in the fuzzy structure, together with the robustness and insensitivity to external effects. Based on this, a unification of Fuzzy–PD, and Fuzzy–PI controllers presented in [Reference Ataşlar-Ayyıldız and Karahan41] is conferred to design a Fuzzy PID surface. As depicted in Figure 2, the proposed controller integrates an FLC with an SMC scheme. The FLC processes error signals and their derivatives (inputs), and its output is combined with an integral term. This combined signal, after scaling with appropriate coefficients, forms the sliding surface output for the SMC. It’s important to note that this design employs integer-order derivatives and integrals. The signal from the equivalent control and the signal from the SMC are summed together to obtain the total control signal.

Figure 2.

Detailed configuration of the FPIDSMC.

As depicted in Figure 2, the FLC unit’s output is determined by the error signal and its derivative. This function is given by:

(22) \begin{equation}u_{FLC}\left(t\right)=f(S_{e}e\left(t\right),S_{ce}de(t)/dt)\end{equation}

Eq. (22) defines scaling factors $S_{e}$ and $S_{ce}$ , responsible for mapping linguistic variables across the discourse universe and scaling the FLC’s error and derivative of error inputs. Both the FLC output and its integral undergo normalization across the universe of discourse using the $S_{PI}$ and $S_{PD}$ scaling factors.

The output value of the FLC and the integral of the output are normalized across the universe of discourse with the $S_{PI}$ and $S_{PD}$ scaling factors. The representation of the fuzzy function is expressed by the nonlinear input-output mapping $f$ function. Consequently, the expression for the SMC sliding surface becomes the following equation [Reference Ataşlar-Ayyıldız and Karahan41].

(23) \begin{equation}u_{FPID}\left(t\right)=S_{PD}u_{FLC}\left(t\right)+S_{PI}d^{-1}u_{FLC}(t)/dt^{-1}\end{equation}

The total control signal $u_{\textit{FPIDSMC}}(t)$ is obtained by combining the equivalent control $u_{eq}(t)$ with the sigmoid function signals, as shown in Eq. (24).

(24) \begin{equation}u_{\textit{FPIDSMC}}\left(t\right)=\omega \frac{u_{FPID}\left(t\right)}{\left| u_{FPID}\left(t\right)\right| +\gamma }+u_{eq}\left(t\right)\end{equation}

Information about how to obtain $u_{eq}(t)$ and the values of $\omega$ and $\gamma$ given in the PIDSMC design.

The FL internal structure of FPIDSMC control uses seven membership functions to represent input and output signals, as indicated in Figure 3. PB and NB are Z-shape membership functions, while the others (NM, NS, ZE, PS, PM) are Gaussian membership functions. The range of membership functions is defined to be [-1, 1] for both inputs and outputs.

Figure 3.

Input/output fuzzy membership functions.

In line with the chosen FL framework, this study employs the Standard Mac Vicar-Whelan Rule Table in [Reference MacVicar-Whelan42]. This rule table, detailed in Table II, defines the FLC’s decision-making logic. As depicted in Figure 4, the corresponding fuzzy control surface provides a visual representation of the rule base. Notably, the rule table exhibits a gradual increase from NB to PB for both inputs and outputs.

Table II.

Fuzzy rule table.

Figure 4.

Visualization of fuzzy inference surface.

3.4. FOFPIDSMC design

In contrast to the integer-order derivative and integral terms employed in FPIDSMC, the proposed FFPIDSMC introduces the concept of FO control. Fractional parameters μ and λ represent the derivative and integral components.

The proposed FOFPID controller is chosen as the sliding surface of the SMC. FOFPIDSMC has been proposed to combine the advantages of the FO PID’s parameter space with the adaptive parameter advantages of FL to increase the efficiency of the SMC. The detailed configuration of FOFPIDSMC is shown in Figure 5.

Figure 5.

Elaborate form of the fractional order fuzzy PID sliding mode control.

The surface view with the fuzzy rule table and input-output membership functions used in FOFPIDSMC is presented in the previous section, as for FPIDSMC.

3.5. Stability analysis using Lyapunov criteria

The stability analysis of the sliding mode controller relies on the Lyapunov stability theorem. In this study, to utilize a positive definite Lyapunov function defined as:

(25) \begin{equation}\mathrm{V}\left(t\right)=\frac{1}{2}\sigma ^{2}(t)\end{equation}

In order for the asymptotic stability condition to be performed in the Lyapunov function, it must be positive ( $\mathrm{V}(t)\gt 0$ ) and its derivative must be negative ( $\dot{\mathrm{V}}(t)\lt 0$ ) at the balance point at the origin $\sigma (t)=0$ .

The expression for the time derivative of the Lyapunov function, $\dot{\mathrm{V}}(t)$ , defined in Eq. (25), is:

(26) \begin{equation}\dot{\mathrm{V}}\left(t\right)=\sigma \left(t\right)\dot{\sigma }\left(t\right)=-\omega \frac{\sigma ^{2}\left(t\right)}{\left| \sigma \left(t\right)\right| +\gamma }\lt 0\end{equation}

the system conditions allow the sliding surface $\sigma (t)=0$ to be reached and maintained.

4.1. Particle Swarm Optimization (PSO) algorithm

This study leverages PSO, a population-based metaheuristic optimization technique improved by Kennedy and Eberhart [Reference Kennedy and Eberhart43] in 1995. Inspired by the collective foraging attitude of bird and fish flocks, PSO is well-suited for optimizing controller parameters. In PSO, a randomly initialized swarm of particles explores the search space, with each particle’s position influenced by its own past experience and the best positions encountered by its neighbors.

Based on this, the top position of every particle and the top position of the particle swarm are stored for later use. In each PSO iteration, the following equations are used to renovate the position and velocity of each particle within the swarm.

(28) \begin{equation}p_{t+1}^{j}=p_{t}^{j}+v_{t+1}^{j}\end{equation}

(29) \begin{equation}v_{t+1}^{j}=\psi \left(\omega _{t}v_{t}^{j}+\zeta _{1}r_{1}\left[p_{t}^{p}-p_{t}^{j}\right]+\zeta _{2}r_{2}\left[p_{t}^{g}-p_{t}^{j}\right]\right)\end{equation}

In the above equations, $t$ represents the iteration number, $p_{t}^{j}$ and $v_{t}^{j}$ are the position and velocity of the $j^{th}$ particle at iteration $t$ , $\psi$ denotes the contraction factor, $\omega _{t}$ serves as the inertia weighting factor influencing the swarm’s searching prowess, $\zeta _{1}$ and $\zeta _{2}$ are the coefficients for learning, and $r_{1}$ and $r_{2}$ are uniformly distributed values between 0 and 1. Additionally, $v_{t+1}^{j}$ (particle $j$ ’s velocity at $t+1$ ), $p_{t+1}^{j}$ (particle $j$ ’s position at $t+1$ ), $p_{t}^{p}$ (particle $j$ ’s personal best position at $t$ ), and $p_{t}^{g}$ (global best position at $t$ ).

4.2. Gray Wolf Optimization (GWO) algorithm

In addition to PSO, the GWO algorithm is utilized for optimization in this work. Proposed by Mirjalili et al. [Reference Mirjalili, Mirjalili and Lewis44], GWO mimics the social hierarchy and hunting behavior of gray wolves. This algorithm incorporates four main wolf types: alpha, beta, delta, and omega. These wolves play distinct roles within the pack, with the alpha acting as the leader, beta wolves providing support, delta wolves fulfilling subordinate roles, and omega wolves occupying the lowest rank.

The mathematical model presented below, inspired by the hunting attitude of gray wolves [Reference Mirjalili, Mirjalili and Lewis44], focuses on the encircling maneuver. This initial stage involves the wolves surrounding the victim, laying the groundwork for the subsequent stages of the hunt.

(30) \begin{equation}\vec {D}=\left| \vec {C}.\vec {X}_{P}\left(t\right)-\vec {X}(t)\right|\end{equation}

(31) \begin{equation}\vec {X}\left(t+1\right)=\vec {X}_{P}\left(t\right)-\vec {A}.\vec {D}\end{equation}

In the above equations, $t$ handles the current iteration. Additionally, $\vec {X}_{P}$ denotes the position of the prey (victim) in the search domain, while $\vec {X}_{P}$ represents the positions of individual wolves within the pack. Furthermore, $\vec {A}$ and $\vec {C}$ are coefficient vectors that influence the hunting behavior and are calculated as follows:

(32) \begin{equation}\vec {A}=\vec {a}.(2.\vec {r}_{1}-1)\end{equation}

(33) \begin{equation}\vec {C}=2.\vec {r}_{2}\end{equation}

where $\vec {r}_{1}$ and $\vec {r}_{2}$ symbolize random vectors uniformly distributed within the range [0, 1]. These random vectors introduce an element of stochasticity into the search process, mimicking the unpredictable nature of hunting behavior. Additionally, $\vec {a}$ signifies a linearly decreasing coefficient vector which starts at a value of 2 and gradually reduces to 0 over iterations, calculated as shown below:

(34) \begin{equation}\vec {a}(t)=2-2t/T\end{equation}

In the context of the GWO algorithm’s optimization process, the following equations model the core hunting strategy inspired by gray wolves. Alpha wolves demonstrate the solution with the best fitness value (best solution), while beta and delta wolves correspond to the second and third-best solutions, respectively. The remaining “wolves” (particles) utilize these leading positions to guide their own movements within the search space, ultimately leading toward optimal solutions.

(35) \begin{equation}\vec {D}_{\propto }=\left| \vec {C}_{1}.\vec {X}_{\propto }-\vec {X}(t)\right|, \vec {D}_{\beta }=\left| \vec {C}_{2}.\vec {X}_{\beta }-\vec {X}(t)\right|, \vec {D}_{\delta }=\left| \vec {C}_{3}.\vec {X}_{\delta }-\vec {X}(t)\right|\end{equation}

(36) \begin{equation}\vec {X}_{1}=\left| \vec {X}_{\alpha }-\vec {a}_{1}\vec {D}_{\alpha }\right|, \vec {X}_{2}=\left| \vec {X}_{\beta }-\vec {a}_{2}\vec {D}_{\beta }\right|, \vec {X}_{3}=\left| \vec {X}_{\delta }-\vec {a}_{3}\vec {D}_{\delta }\right|\end{equation}

(37) \begin{equation}\vec {X}_{P}\left(t+1\right)=\left(\vec {X}_{1}+\vec {X}_{2}+\vec {X}_{3}\right)/3\end{equation}

In these equations of hunting strategy, $\vec {C}_{1}$ , $\vec {C}_{2}$ , $\vec {C}_{3}$ , $\vec {a}_{1}$ , $\vec {a}_{2}$ , $\vec {a}_{3}$ express random vectors. $\vec {X}_{\alpha }$ , $\vec {X}_{\beta }$ and $\vec {X}_{\delta }$ indicate the location of alpha, beta and delta, respectively. $\vec {X}_{1}$ , $\vec {X}_{2}$ , and $\vec {X}_{3}$ determine the notional status of alpha, beta and delta, respectively. $\vec {X}_{P}(t+1)$ is defined as the new position of the victim.

4.3. Hybrid algorithm based on GWO and PSO

Metaheuristic methods are used to solve problems by searching for the best solution, which can be either local or global. While metaheuristic techniques do not assure the global optimum, they can maintain accomplished solutions to complex problems in a relatively simple and efficient manner [Reference Nazari-Heris, Mohammadi-Ivatloo and Gharehpetian45]. Researchers sometimes hybridize metaheuristics to improve their search capabilities and combine their advantages [Reference Ting, Yang, Cheng, Huang and Yang46]. Based on all these explanations, exploitation, and exploration capabilities are important locally and globally [Reference Devarapalli, Bhattacharyya and Sinha47]. The PSO algorithm’s established success in diverse optimization problems stems from its minimal parameter requirements and rapid convergence rate, as demonstrated in [Reference Ting, Yang, Cheng, Huang and Yang46]. This well-proven effectiveness across various fields makes it a compelling choice for inclusion in this hybrid optimization study. It is noteworthy that the GWO algorithm has a very high exploitation ability (local optimum success) but a weak exploration ability (global optimum) [Reference Karahan and Karci35]. However, effective optimization requires a balance between exploration and exploitation. While exploration allows for diverse exploration of the search space, exploitation focuses on refining solutions within promising regions, as emphasized in ref. [Reference Ting, Yang, Cheng, Huang and Yang46]. To increase this disadvantage of GWO, the PSO algorithm is implemented to update the positions of search agents within the GWO algorithm by hybridizing it with the well-known PSO algorithm. Thus, instead of searching agents going to a random location, which causes them to get stuck in local minima in GWO, it is aimed to find global minima by using the discovery feature of PSO.

Hybridization has been proposed to address the disadvantages and combine the superiorities of the GWO and PSO algorithms by utilizing the GWO’s exploitation capability and the PSO’s exploration capability [Reference Karahan and Karci35, Reference Zhang, Lin, Mao, Liu, Dou and Liu48–Reference Gupta and Deep51]. Thus, the fundamental process steps of the Hybrid GWO-PSO technique, which is a new idea, are detailed in ref. [Reference Shaheen, Hasanien and Alkuhayli52]. According to this, the key steps involve initializing search agents, updating positions using GWO, identifying the best agents, switching to PSO, and iteratively refining the solution. The approach aims to achieve optimal results while ensuring stability. The technique is visually depicted in Figure 7.

Figure 7.

Hybrid gray wolf optimization and particle swarm optimization algorithm flowchart.

To provide a clearer understanding of the hybridization process depicted in Figure 7, the algorithm integrates the GWO and PSO mechanisms as follows: Initially, the search agents’ positions are updated using the GWO algorithm’s hierarchical leadership structure, where alpha, beta, and delta wolves guide the rest of the agents toward promising regions based on their exploitation capability. After this stage, the top-performing agents identified by the GWO are further refined using the PSO algorithm’s velocity and position update rules.

Specifically, the PSO algorithm introduces an exploration mechanism by updating the positions of selected agents according to their previous velocities, their best-known positions (personal best), and the global best position achieved so far. This dual mechanism ensures that the search agents do not stagnate in local optima, as the PSO introduces a stochastic component to encourage diverse exploration of the search space. The iterative process alternates between GWO and PSO updates, leveraging the complementary strengths of both algorithms to balance exploitation and exploration effectively.

This detailed step-by-step integration is visually summarized in Figure 7 to illustrate the dynamic interaction between the two algorithms in the hybrid framework.

In this study, the number of agents for all algorithms used in the optimization process is set to 30, the number of iterations is set to 100, and the number of runs is set to 10. The PSO algorithm’s inertia weighting factor ( $\omega$ ) is set to 0.8, the contraction factor ( $\psi$ ) is set to 0.6, the learning factors ( $\zeta _{1}$ , $\zeta _{2}$ ) are set to 2, and the coefficients ( $r_{1}$ and $r_{2}$ ) are randomly chosen in the range $[0,1]$ . The GWO algorithm’s coefficients ( $\vec {r}_{1}$ and $\vec {r}_{2}$ ) are randomly selected in the range $[0,1]$ and the linear vector ( $\vec {a}$ ) decreases from 2 to 0 in each iteration.

5.1. Dynamic performance analysis

Transient and steady-state responses for the 6R UR5 robot manipulator system controlled by PIDSMC, FOPIDSMC, FPIDSMC, and FOFPIDSMC are analyzed based on the objective function given in Eq. (27) while the reference input is a unit step. The optimal controller parameters are acquired by handling the objective function presented in Eq. (27) and optimizing it with GWO-PSO.

As a consequence, all simulations are conducted using GWO-PSO-based optimization for different scenarios with optimized controllers, as the parameter values of all joints of UR5 of each controller are indicated in Table III. The optimization is run 10 times for each controller, and the results are averaged to obtain the parameter values indicated in Table III.

Table III.

Optimal controller parameters for 6R UR5 robot manipulator.

Table IV.

Dynamic response comparison of controllers for the 6R UR5 robot manipulator system.

The dynamic responses of the optimized controllers for all joints in terms of rise time ( $t_{r}$ ), settling time ( $t_{s}$ ), overshoot ( $M_{p}$ ), and state error ( $e_{ss}$ ) stability are given in Table IV. Additionally, in this table, the mean absolute error (MAE) criterion given in Eq. (38) has been used as the objective function to evaluate the dynamic performance.

(38) \begin{equation}e_{{MAE_{i}}}=\frac{1}{T}\sum _{t=1}^{T}\left| e_{i}(t)\right|\end{equation}

where $e_{i}(t)$ is the trajectory error of $t$ th sample of $i$ th joint and $T$ the number of sample.

As can be seen from Table IV, it can be concluded that the proffered GWO-PSO-based FOFPIDSMC with parameter tuning exhibits superior dynamic performance with the quickest settling time, minimal steady-state error, and MAE values for all joints of the robot. If evaluated in terms of rise times, it seems that FPIDSMC and FOFPIDSMC controllers give the lowest values than other controllers for all joints. In addition to these analyses, it is observed that although the percentage over-shot values of FL PIDSMC are higher than non-FL PIDSMC, FL PIDSMC performs better than other controllers in terms of transient dynamic responses according to other performance.

Figure 8 illustrates the sliding surfaces and tracking errors obtained for each joint using the proffered sliding mode controllers. It is evident that the suggested controllers achieve finite-time convergence and robustness of the six-joint tracking error. It is observed that for all joints, FOFPIDSMC has better performance with lower tracking errors than other controllers. A sliding mode is achieved on the sliding surface and reaches zero in finite time. Although similar behavior is observed on sliding surfaces, there are numerical differences due to the joint coupling effect depending on the dynamic structure of the joint.

Figure 8.

Sliding surfaces (a) and trajectory tracking errors (b) for different control methodologies.

5.2. Controller performance analysis under different trajectory tracking

The precision of robot manipulator closed-loop control systems is examined as the end effector follows the rectangular path in Cartesian space shown in Figure 9 as the position reference. The position graphs of all joints in the joint space while the end effector of UR5 follows the rectangular trajectory in Cartesian space are presented in Figure 10. From Figure 10, it can be noticed that for all joints, FOFPIDSMC has a quicker and more precise system reaction compared to the others. Furthermore, it has been determined that according to Figure 10, the tracking accomplishment of the FOFPIDSMC controller adjusted by GWO-PSO is excellent to other controllers. This detection shows that the suggested FOFPIDSMC controller provides a preferable disturbance rejection skill in case of sudden set point replacement.

Figure 9.

The appearance of UR5 in MATLAB with the trajectory in Cartesian space.

Figure 10.

The performance of controllers for end effector rectangular trajectory tracking.

In addition, in order to show the control effort exerted by the controllers the RMS (Root Mean Square) of the control torque can be calculated as follows:

(39) \begin{equation}U_{{RMS_{i}}}=\sqrt{\frac{1}{T}\sum _{t=1}^{T}({u_{i}}(\mathrm{t}))^{2}}\end{equation}

where $U_{{RMS_{i}}}$ is RMS value of control signal of $i$ th joint, and $u_{i}(t)$ is control signal of $t$ th sample of $i$ th joint. In this context, the MAE values of each joint and RMS values of the torques of each joint for the tracking efficiency of the controllers mentioned in this study are presented in Figure 11.

Figure 11.

MAE values of each joint (a) and RMS values of the torques of each joint (b) for the tracking performance.

Considering the MAE values for each joint in Figure 11a, the tracking efficiency of the FOFPIDSMC controller compared to PIDSMC, significant improvements of 92.06% in the first joint, 79.48% in the second joint, 95.75% in the third joint, 84% in the fourth joint, 84.32% in the fifth joint, and finally 81.32% in the sixth joint were achieved, respectively. Likewise, in terms of torque RMS values of the joints, when FOFPIDSMC and PIDSMC are compared, a 25.44% performance improvement is observed in the average values of torque RMS of all joints.

This overall performance increase in torque RMS means less energy consumption during trajectory tracking. In addition, in terms of MAE, it has been revealed that the system with FOFPIDSMC controller followed the desired trajectory with less error than the other controllers (PIDSMC, FOPIDSMC, FPIDSMC), and it resulted in an improvement in the operating performance of the robot manipulator with low RMS values.

5.3. Controller performance analysis under disturbance

The main target of a robust control algorithm is to satisfactorily prevent undesirable disturbances by ensuring proper tracking of the reference trajectory. Therefore, in this study, during trajectory tracking, an external disturbance effect is applied to all joints of the robot manipulator between the controller output and the robot system for a certain period of time, with the pulse function given in Eq. (40).

(40) \begin{equation}\sigma _{d}\left(t\right)={\Gamma} \left[u\left(t-a\right)-u(t-b)\right]\end{equation}

${\Gamma}$ given in the above equation refers to the coefficient of the disturbance effect, and $a$ and $b$ refer to the start and end times of the disturbance effect.

Two different scenarios of instantaneous distortion, 0.2 s (scenario 1), and 2 s (scenario 2), were produced and applied to the base, shoulder, and elbow joints with a pulse size of 100 N-m, and three joints forming the wrist with a pulse size of 20 N-m.

The impact of external disturbances applied to the controller output and system input while the controllers track the reference trajectory was investigated in detail for controllers adjusted by GWO-PSO. A full analysis is performed, where the averages of the MAE and torque RMS values obtained for all joints are presented graphically in Figure 12.

Figure 12.

Averages of MAE (a) and torque RMS (b) values of all joints in disturbance rejection scenarios.

A trajectory tracking task and the Cartesian-XY position curve of the endpoint for this case are demonstrated in Figure 13 where it can be clearly observed that the FOFPIDSMC controller follows a curve closer to the reference than the other controller. After examining the obtained MAE and torque RMS values, it can be redounded that the FOFPIDSMC controller outperforms than others. From these results, it can be concluded that for scenario 1, the FOFPIDSMC controller can improve the average MAE values of all joints by 88.6%, 72.77%, and 57.42% compared to the PIDSMC, FOPIDSMC, and FPIDSMC controllers, respectively.

Figure 13.

Scenario 1 (a) and 2 (b) trajectory tracking mission and scenario 1 (c) and 2 (d) Cartesian-XY position curve of the endpoint.

Moreover, for scenario 2, it can be issued that the FOFPIDSMC controller makes better the average MAE values of all joints by 94.67%, 58.21%, and 27.74% compared to the PIDSMC, FOPIDSMC, and FPIDSMC controllers, respectively. For another performance analysis in the mentioned disturbance scenarios, the average of the RMS values of all joints was compared. In this context, it can be observed that for scenario 1, the FOFPIDSMC controller is at a better level by 89.87%, 72.26%, and 47.7% compared to the PIDSMC, FOPIDSMC, and FPIDSMC controllers, respectively. Moreover, in scenario 2, it can be understood that the FOFPIDSMC controller controls better by 95.39%, 57.97%, and 25.46% compared to the PIDSMC, FOPIDSMC, and FPIDSMC controllers, respectively.

The proposed FOFPIDSMC controller exhibits better settling time and tracking performance relative to other controllers, as evident in Figure 13(c-d). Additionally, torque values play a crucial role in determining system energy consumption. Figure 13(a-b) demonstrates that the FOFPIDSMC controller system produces lower torque values than the others, indicating reduced energy expenditure. Situated on these findings, it can be concluded that the FOFPIDSMC controller system provides enhanced tracking performance with minimal torque under external disturbances.

5.4. Controller performance analysis under uncertainty in mass

In another test of the controllers’ robustness, the mass of each link of the robot manipulator was increased to three times. For this purpose, the performance of the UR5’s end effector in tracking the parallel circle trajectory with respect to the Cartesian-XY plane of the main coordinate system was examined, as shown in Figure 14. In this trajectory tracking, it is clearly understood from Figure 15 that the second and fourth joints of the robot are more affected by the mass change due to gravity. Moreover, the FOFPIDSMC controller based on GWO-PSO performs better than other controllers from the point of convergence and steady-state error.

Figure 14.

Robot manipulator and Cartesian-XY position curve of the endpoint.

Figure 15.

The performance of controllers under the mass change for trajectory tracking.

Figure 16 graphically shows the MAE and torque RMS values, where improvement is easily observed for each joint compared to the other controllers (PIDSMC, FOPIDSMC, FPIDSMC). Accordingly, improvements of 99.05%, 98.01%, 98.26%, 97.54%, 95.51%, and 94.83% were obtained in MAE values for the 1st, 2nd, 3rd, 4th, 5th, and 6th joints from the PIDSMC controller to the FOFPIDSMC side. In addition, improvements of 9.27%, 8.52%, 64.73%, 0.65%, and 16.9% were observed in torque RMS values for the 1st, 2nd, 4th, 6th joints, and average from the PIDSMC controller to the FOFPIDSMC side. However, even though there is a deterioration of 8.87% and 5.12% in torque RMS values in only the 3rd and 5th joints, FOFPIDSMC is superior in overall improvement. Based upon the mass change analysis, it can be said that the FOFPIDSMC controller outperforms the other controllers mentioned in this study.

Figure 16.

MAE values of each joint (a) and RMS values of the torques of each joint (b) for the tracking performance under mass change.

The comprehensive comparative analysis of the proposed FOFPIDSMC controller with PIDSMC, FOPIDSMC, and FPIDSMC controllers has demonstrated significant improvements in control performance. However, it is crucial to discuss the computational complexity and real-time applicability of the proposed controller to provide additional context for its practical utility. The FOFPIDSMC controller integrates FO control and FL, which increases its computational demands compared to traditional PIDSMC, FOPIDSMC, and FPIDSMC controllers. The calculation of fractional derivatives and the fuzzy inference process require more processing power and memory. However, advancements in computational hardware and optimization techniques can mitigate these challenges.

Implementing the FOFPIDSMC controller in real-time systems presents several challenges. The increased computational load may affect the controller’s responsiveness in real-time applications. Ensuring real-time performance would require efficient algorithm optimization and possibly dedicated hardware to handle the computations without compromising the control system’s responsiveness. Despite these challenges, the proposed controller offers superior performance in terms of accuracy and robustness. Future work will focus on optimizing the algorithm for real-time applications and testing the controller on physical robotic systems to validate its practical utility and address any implementation challenges.