Optimized Adaptive Fuzzy Synergetic Controller for Suspended Cable-Driven Parallel Robots

Abstract

A suspended cable-driven parallel robot is a type of lightweight large-span parallel robot. The stability and control of this multi-input multi-output robot are studied in this work to overcome its inherited vulnerability to disturbance. An adaptive fuzzy synergetic controller is proposed to overcome these issues, combining synergetic control theory with adaptive fuzzy logic to ensure robust trajectory tracking. The parameters of the controller are optimized using the Dragonfly Algorithm, a metaheuristic technique known for its simplicity and fast convergence. The adaptive fuzzy synergetic controller is tested on a suspended cable-driven parallel robot model under both disturbance-free and disturbed conditions, demonstrating global asymptotic stability and superior tracking accuracy compared to existing controllers. Simulation results show the proposed controller achieves minimal tracking error and improved robustness in the presence of dynamic uncertainties, validating its practical applicability in industrial scenarios. The findings highlight the effectiveness of integrating synergetic control, fuzzy logic adaptation, and optimization for enhancing the performance and reliability of suspended cable-driven parallel robots.

1. Introduction

A cable-driven parallel robot (CDPR) is a type of parallel robot where the end-effector (EE) is supported by multiple cables and moved by motor actuators mounted on a fixed frame. Unlike traditional parallel robots that use rigid links, CDPRs employ flexible cables to control the EE’s position and orientation. This design offers several advantages, including a higher payload-to-weight ratio, lower inertia, and the capability to achieve high speeds and accelerations. Additionally, the use of flexible cables allows CDPRs to achieve the largest workspace among all parallel robots. However, one major drawback of CDPRs is that the cables can only transmit positive tension forces, as they lack the ability to resist compression. This inherent limitation poses significant challenges for control design, especially under disturbances or uncertainties.

CDPRs are typically classified based on the number of cables and their degrees of freedom into under-constrained, fully constrained, and redundantly constrained systems. They can also be categorized by their operational dimensions into planar and spatial configurations. Furthermore, CDPRs are divided into suspended and non-suspended types depending on the cable arrangement relative to the EE. Suspended CDPRs, where all drive cables are positioned above the EE and gravity serves as a virtual cable, are widely used for large-load operations. This configuration provides advantages such as a reduced likelihood of cable collisions and efficient distribution of payload forces across the cables. However, suspended CDPRs face critical challenges, including low vertical rigidity, and susceptibility to external disturbances, making their control significantly more complex than that of non-suspended CDPRs [1,2,3].

The control of suspended spatial CDPRs is an active area of research due to their structural sensitivity to external disturbances, uncertainties in model parameters, and nonlinear dynamics. To ensure stable and precise control, robust and adaptive control strategies are necessary. For instance, previous studies have explored linear algorithms and sliding mode control, demonstrating stability under varying conditions [4]. Additionally, adaptive synchronization control has been proposed to tackle kinematic and dynamic uncertainties [5], while robust adaptive fuzzy controllers employing sliding mode and fuzzy logic approximation have been used for trajectory tracking [6]. Furthermore, techniques such as fuzzy PID controllers optimized with metaheuristic algorithms, like whale optimization, have also been applied to enhance control robustness and accuracy [7]. Despite these advancements, issues such as chattering, computational complexity, and sensitivity to parameter variations remain.

Synergetic control theory, introduced by [8], offers a promising solution to these challenges. This nonlinear control strategy leverages invariance principles similar to sliding mode control but avoids its inherent chattering issues. Synergetic control synthesizes the nonlinear dynamics of the system, forcing it to operate on a predefined manifold, thereby reducing the system’s order and enhancing robustness [9,10,11]. While effective, the performance of synergetic control depends on precise parameter tuning, system parameters accuracy, and absence of disturbances. To address the first one, optimization techniques, especially metaheuristic optimizations, can be employed to improve control performance. For the second and third ones, adding an adaptive controller and compensator is one of the effective solutions. Fuzzy neural networks are one of the best tools that provide adaptability and approximation power inherited by fuzzy logic and neural networks capabilities. In the past decade, there were several works that utilized fuzzy neural network adaptive synergetic controllers to deal with Single-Input Single-Output (SISO) nonlinear systems. A robust controller for nonlinear systems, leveraging adaptive fuzzy basis systems and synergetic control, has been introduced by [12]. The same technique has been utilized for hysteresis nonlinear systems [13], uncertain discrete-time nonlinear dynamic systems [14], a power system stabilizer [11,15], thermoelectric generators power extraction enhancement [16], and voltage control in DC/DC converters [17]. In [18], in addition to the adaptive fuzzy synergetic controller, particle swarm optimization (PSO) was used to optimize the gains of synergetic controllers that have been used to control a power system stabilizer. While the adaptive fuzzy synergetic controller has been extensively applied to SISO systems, its application to multi-input multi-output (MIMO) systems, such as CDPRs, remains unexplored, presenting a significant gap in the research landscape. Also, employing more recent optimization techniques offers significant improvements over previous works. The Dragonfly Algorithm (DA) has emerged as an efficient and straightforward metaheuristic optimization method. DA mimics the social behavior of dragonflies during foraging and migration, offering advantages such as faster convergence and fewer tuning parameters compared to traditional methods [19].

In this work, an adaptive fuzzy synergetic controller (AFSC) is proposed, adapting the existing adaptive fuzzy synergetic control framework, for controlling a suspended CDPR incorporating unique modifications to address the specific challenges of this system. The controller combines the robust properties of synergetic control with the adaptive approximation capabilities of a fuzzy basis system to compensate for disturbances and uncertainties. Additionally, the DA is used to optimize controller parameters, ensuring improved performance and stability. The proposed controller’s effectiveness is validated through simulation, demonstrating superior tracking accuracy and robustness compared to existing methods, such as adaptive fuzzy sliding mode controllers [6]. By addressing challenges in parameter uncertainties, external disturbances, and control optimization, this study contributes to advancing the robust control of suspended CDPRs for industrial applications.

The remainder of this paper is organized as follows: Section 2 presents the kinematic and dynamic modeling of the suspended CDPR system, providing the foundational equations used for control design. Section 3 details the proposed optimized AFSC, including the synergetic control framework, fuzzy basis system, and parameter optimization using DA. Section 4 presents simulation results, comparing the performance of the proposed controller with previous methods under both ideal and disturbed conditions. Finally, Section 5 concludes the paper with key findings, implications for practical applications, and directions for future work.

2. Kinematic and Dynamic Modeling of a Suspended CDPR

The basic diagram of the system model is shown in Figure 1. The workspace is encircled by a cuboid frame, where each upper corner is topped with a cable pulley. At the base of the frame, four actuators equipped with winches are placed. The cables, which transmit tension from the actuators to the EE, are guided by these winches and pulleys. The origin $O\left(\mathrm{0,0},0\right)$ is positioned at one corner of the base. The EE’s location is denoted by $P(x,y,z)$. The top plane’s corners ($A_i(x_{Ai},y_{Ai},z_{Ai}), i=\mathrm{1,2},\mathrm{3,4}$) mark the pulleys’ locations and serve as the cables’ anchor points. The cable lengths are represented by $L_i, i=\mathrm{1,2},\mathrm{3,4}$, and the frame’s dimensions are denoted by $d_j, j=\mathrm{1,2},3$.

2.1. Kinematic Modeling

Employing the loop closure technique for the $i^{th}$ cable results in the corresponding inverse geometric transformation [20,21]:

$$l_i{\mathit{u}}_i=\mathit{P}-{\mathit{A}}_i, i=\mathrm{1,2},\mathrm{3,4},$$

(1)

where $l_i$ and ${\mathit{u}}_{\mathit{i}}$ ∈ ℝ⁴ are the cable length and the unit vector of the $i^{th}$ cable vector, respectively. $\mathit{P}={[x y z]}^T$, and ${\mathit{A}}_i={\left[x_{Ai},y_{Ai},z_{Ai}\right]}^T$. $l_i$ and ${\mathit{u}}_{\mathit{i}}$ are defined as:

$$l_i=\Vert \mathit{P}-{\mathit{A}}_{\mathit{i}}\Vert , {\mathit{u}}_{\mathit{i}}={\displaystyle \frac{\mathit{P}-{\mathit{A}}_{\mathit{i}}}{l_i}} i=\mathrm{1,2},\mathrm{3,4}$$

(2)

where ||∙|| is the Euclidean norm. Multiplying the time derivative of (1) by (u_i)^T gives:

$$l_i\dot{{\mathit{u}}_i}{{\mathit{u}}_i}^T+\dot{l_i}{\mathit{u}}_i{{\mathit{u}}_i}^T=\dot{\mathit{P}}{{\mathit{u}}_i}^T, i=\mathrm{1,2},\mathrm{3,4}$$

(3)

The value of $\dot{{\mathit{u}}_i}{{\mathit{u}}_i}^T$ is zero which leads to:

$$\dot{l_i}={{\mathit{u}}_i}^T\dot{\mathit{P}}, i=\mathrm{1,2},\mathrm{3,4}.$$

(4)

Equation (4) can be written in a concise form as:

$$\dot{\mathit{L}}=\left[\begin{array}{ccc}{\displaystyle \frac{x-x_{A1}}{l_1}}& {\displaystyle \frac{y-y_{A1}}{l_1}}& {\displaystyle \frac{z-z_{A1}}{l_1}}\\ {\displaystyle \frac{x-x_{A2}}{l_2}}& {\displaystyle \frac{y-y_{A2}}{l_2}}& {\displaystyle \frac{z-z_{A2}}{l_2}}\\ {\displaystyle \frac{x-x_{A3}}{l_3}}& {\displaystyle \frac{y-y_{A3}}{l_3}}& {\displaystyle \frac{z-z_{A3}}{l_3}}\\ {\displaystyle \frac{x-x_{A4}}{l_4}}& {\displaystyle \frac{y-y_{A4}}{l_4}}& {\displaystyle \frac{z-z_{A4}}{l_4}}\end{array}\right]\dot{\mathit{P}}=\mathit{J}\dot{\mathit{P}}$$

(5)

where $\mathit{J}$ represents the Jacobian matrix of the system.

2.2. Dynamic Modeling

The dynamic model of the suspended CDPR is derived using the Euler–Lagrange formulation, which considers the system’s kinetic and potential energy to compute the equations of motion. The Lagrangian is defined as $\mathcal{L}=\mathit{\kappa }-\mathit{\rho },$ where $\mathit{\kappa }$ and ρ are the kinetic and potential energies of the system, respectively [22]. In Cartesian coordinates of the EE, $\mathit{\kappa }$ and ρ are defined as:

$$\mathit{\kappa }={\displaystyle \frac{1}{2}}\dot{{\mathit{P}}^{\mathit{T}}}\mathit{M}\dot{\mathit{P}} \text{and} \mathit{\rho }={\mathit{G}}^{\mathit{T}}\mathit{P},$$

(6)

where $\mathit{M}=m{\mathit{I}}_{3\times 3}, \mathit{G}={\left[\begin{array}{ccc}0& 0& -mg\end{array}\right]}^T,$m is the EE mass, and g is the acceleration of gravity.

The Lagrangian equation of the EE is:

$$\mathit{\tau }={\displaystyle \frac{d}{dt}}{\displaystyle \frac{\mathsf{\partial }\mathcal{L}}{\mathsf{\partial }\dot{\mathit{P}}}}-{\displaystyle \frac{\mathsf{\partial }\mathcal{L}}{\mathsf{\partial }\mathit{P}}}=\mathit{M}\ddot{\mathit{P}}+\mathit{G}$$

(7)

where $\mathit{\tau }$ ∈ ℝ³ is the generalized force vector in task space.

To transform the forces from the task space to the joint space,$\mathit{\tau }$ can be expressed as [23]:

$$\mathit{\tau }={\mathit{J}}^T\mathit{T}.$$

(8)

where T ∈ ℝ⁴ is the tension vector in the joint space. Now, (7) can be written as

$${\mathit{J}}^T\mathit{T}=\mathit{M}\ddot{\mathit{P}}+\mathit{G}+\mathit{\delta },$$

(9)

where $\mathit{\delta }$ accounts for all the disturbances and uncertainties in the system and it is assumed to be bounded by a scalar value δ, i.e., $‖\mathit{\delta }‖\le \delta$.

Table 1. Kinematic and dynamic variables of the suspended CDPR.

Parameter	Symbol	Description
Frame Dimensions	$d_1{, d}_2,d_3$	Length, width, and height of the frame.
Top plane’s corners	$A_1{, A}_2,A_3{,A}_4$	Positions of the top corners of the frame.
EE position	$\mathit{P}={[x,y,z]}^T$	position of the EE.
Cable length	$\mathit{L}={[l_1{, l}_2,l_3,l_4]}^T$	length of the cables.
EE mass matrix	$\mathit{M}=m{\mathit{I}}_{3\times 3}$	m is the mass of EE, and I is the identity matrix.
EE gravity vector	$\mathit{G}={[0,0,-mg]}^T$	$g=9.81 \mathrm{m}/{\mathrm{s}}^2.$
Generalized force vector	$\mathit{\tau }={[{\tau }_1,{\tau }_2,{\tau }_3]}^T$	Force in task space.
Tension vector	$\mathit{T}={[T_1,T_2,T_3,T_4]}^T$	Force in joint space.
Jacobian matrix	$\mathit{J}\in {\mathbb{R}}^{4\times 3}$	Transformation matrix from joint space to task space.
Disturbances and uncertainties component vector	$\mathit{\delta }={[{\delta }_1,{\delta }_2,{\delta }_3]}^T$	bounded value.

summarizes the variables used above.

3. Controller Design

3.1. Synergetic Controller

The objective of synthesizing a synergetic controller is to force the system to operate on a certain manifold $\mathit{\sigma }$ = 0. The macro-variable of the manifold equation is defined as [15,17]:

$$\mathit{\sigma }=\dot{\mathit{e}}+\mathit{v}\mathit{e}$$

(10)

where $\mathit{e}={\mathit{P}}_{\mathit{d}}-\mathit{P}$, $\mathit{v}$ ∈ ℝ^3×3 is a constant positive definite diagonal matrix, and ${\mathit{P}}_{\mathit{d}}$ is the desired position trajectory.

The desired constraint function that dictates the dynamic evolution of the macro-variable is defined as:

$$\mathit{s}=\mathit{\sigma }+\mathit{w}\dot{\mathit{\sigma }}=0$$

(11)

$$\mathit{\sigma }+\mathit{w}(\ddot{\mathit{e}}+\mathit{v}\dot{\mathit{e}})=0$$

(12)

$$\mathit{\sigma }+\mathit{w}{\ddot{\mathit{P}}}_d-\mathit{w}{\mathit{M}}^{-1}\mathit{\tau }+\mathit{w}{\mathit{M}}^{-1}\mathit{G}+\mathit{w}\mathit{v}\dot{\mathit{e}}=0,$$

(13)

where w ∈ ℝ^3×3 is a constant positive definite diagonal matrix that dictates the rate of convergence toward the manifold $\mathit{\sigma }$ = 0.

Then, the control law can be written as:

$$\mathit{\tau }=\mathit{M}{\mathit{w}}^{-1}\mathit{\sigma }+\mathit{M}{\ddot{\mathit{P}}}_d+\mathit{G}+\mathit{M}\mathit{v}\dot{\mathit{e}}.$$

(14)

The candidate Lyapunov function for checking the stability of the control law is chosen to be a positive definite of the form:

$$V={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\sigma }}^{\mathit{T}}\mathit{M}\mathit{\sigma }$$

(15)

Taking the time derivative of (15) and compensating with (11) gives

$$\dot{V}=-{\displaystyle \frac{\mathbf{1}}{\mathit{w}}}{\mathit{\sigma }}^{\mathit{T}}\mathit{M}\mathit{\sigma }.$$

(16)

That is, the control law of (14) guarantees the system stability [24].

3.2. Fuzzy Basis System

Fuzzy logic systems employ IF–THEN rules to convert human expertise into automatic control strategies. To ensure reliable performance amid significant uncertainty or changes in plant parameters and structures, these systems must be adaptive. Although fuzzy logic is a powerful approximation tool, it lacks inherent adaptability. Combining fuzzy logic with adaptation methods such as neural networks harnesses the strengths of both approaches [25,26]. By setting certain parameters as flexible, an adaptive fuzzy system can be conceptualized as a specific type of neural network. The fuzzy basis system is one of the simplest methods to achieve this [27].

A fuzzy logic system with multiple inputs and outputs carries out a mapping from fuzzy sets in x ∈ ℝ^M to fuzzy sets in y ∈ ℝ^N. The system employs a center-average defuzzifier, a product inference, a singleton fuzzifier, and Gaussian membership functions (MFs). For $x_m,m=\mathrm{1,2} \dots M$ inputs, $y_n,n=\mathrm{1,2} \dots N$ outputs, R rules, ${\mu }_{F_m^r}$ input MFs, and ${\mu }_{G_n^r}$ output MFs (r = 1,2 … R), the $n^{th}$ fuzzy output can be expressed as [6,28]:

$$y_n={\displaystyle \frac{{\sum }_{r=1}^R{\mu }_{G_n^r}\left({\prod }_{m=1}^M{\mu }_{F_m^r}\left(x_m\right)\right)}{{\sum }_{r=1}^R\left({\prod }_{m=1}^M{\mu }_{F_m^r}\left(x_m\right)\right)}}$$

(17)

The fuzzy basis function can be defined as:

$$z_r(x)={\displaystyle \frac{{\prod }_{m=1}^M{\mu }_{F_m^r}\left(x_m\right)}{{\sum }_{r=1}^R\left({\prod }_{m=1}^M{\mu }_{F_m^r}\left(x_m\right)\right)}}$$

(18)

As a result, (17) can be written as follows:

$$y_n={\mathit{\phi }}_n^T\mathit{z}\left(\mathit{x}\right)$$

(19)

where $\mathit{z}\left(x\right)={\left[z_1\left(\mathit{x}\right),z_2\left(\mathit{x}\right)\dots z_R\left(\mathit{x}\right)\right]}^T$ is the fuzzy basis function vector, $\mathit{x}={\left[\mathit{P}\dot{\mathit{P}}\right]}^T$, and ${\phi }_n^T=\left[{\mu }_{G_n^1}, {\mu }_{G_n^2}\dots {\mu }_{G_n^R}\right]$ denotes the output MFs. Output MFs can be chosen to be the centers of the fuzzy subset (Singleton) for simplicity. ${\phi }_n^T$ is considered the adaptive weights of the fuzzy basis system.

Furthermore, the overall output of a MIMO fuzzy basis system can be represented as:

$$\mathit{y}={\mathit{\phi }}^T\mathit{z}\left(\mathit{x}\right)$$

(20)

3.3. AFSC

The proposed synergetic controller of Equation (14) can be expressed by the following equation:

$$\mathit{\tau }=\mathit{M}{\mathit{w}}^{-1}\mathsf{\sigma }+\mathit{M}{\ddot{\mathit{P}}}_d+\mathit{G}+\mathit{M}\mathit{v}\dot{\mathit{e}}+\widehat{\mathit{F}}(\mathit{x}|\tilde{\mathit{\phi }})$$

(21)

where $\widehat{\mathit{F}}\left(\mathit{x}|\tilde{\mathit{\phi }}\right)={\mathit{\phi }}^T\mathit{z}\left(\mathit{x}\right)$ is the adaptive fuzzy logic compensation control for the composite disturbance $\mathit{\delta }$ of Equation (13) and as shown in Figure 2.

The stability of the hybrid controller of Equation (21) can be verified by using Lyapunov theory and the candidate Lyapunov function is defined as:

$$V={\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}({\mathsf{\sigma }}^{\mathit{T}}\mathit{M}\mathsf{\sigma }+{\tilde{\mathit{\phi }}}^T\mathsf{\Lambda }\tilde{\mathit{\phi }})$$

(22)

where $\tilde{\mathit{\phi }}=\mathit{\phi }-{\mathit{\phi }}^{\mathrm{*}},$ ${\mathit{\phi }}^{\mathrm{*}}$ is the ideal fuzzy parameters and $\mathsf{\Lambda }$ is a positive definite diagonal matrix.

The time derivative of (22) is derived as:

$$\dot{V}={\mathsf{\sigma }}^{\mathit{T}}\mathit{M}\dot{\mathsf{\sigma }}+{\tilde{\mathit{\phi }}}^T\mathsf{\Lambda }\dot{\tilde{\mathit{\phi }}}$$

(23)

Compensation for Equations (9), (10), and (21) yields to:

$$\dot{V}={-\mathsf{\sigma }}^{\mathit{T}}\left(\mathit{M}{\mathit{w}}^{-\mathbf{1}}\mathsf{\sigma } + \mathit{\delta } - \widehat{\mathit{F}}\left(\mathit{x}|\tilde{\mathit{\phi }}\right)\right) + {\tilde{\mathit{\phi }}}^T\mathsf{\Lambda }\dot{\tilde{\mathit{\phi }}}$$

(24)

$$\dot{V}={-\mathsf{\sigma }}^{\mathit{T}}\mathit{M}{\mathit{w}}^{-\mathbf{1}}\mathsf{\sigma }{ - \mathsf{\sigma }}^{\mathit{T}}{\mathit{e}}_{\mathit{a}\mathit{p}\mathit{r}\mathit{x}} + {\tilde{\mathit{\phi }}}^T\mathsf{\Lambda }\dot{\tilde{\mathit{\phi }}}{{ - \mathsf{\sigma }}^{\mathit{T}}\mathit{\phi }}^T\mathit{z}(\mathit{x})$$

(25)

where ${\mathit{e}}_{\mathit{a}\mathit{p}\mathit{r}\mathit{x}}=\mathit{\delta }-\widehat{\mathit{F}}\left(\mathit{x}|\tilde{\mathit{\phi }}\right)$ is the approximation error. The minimum approximation error ${\mathit{e}}_{\mathit{a}\mathit{p}\mathit{r}\mathit{x}}$ can be made negligibly small by designing the adaptive fuzzy logic system with an adequate number of rules. By satisfying ${\tilde{\mathit{\phi }}}^T\mathsf{\Lambda }\dot{\tilde{\mathit{\phi }}}{{-\mathsf{\sigma }}^{\mathit{T}}\mathit{\phi }}^T\mathit{z}\left(\mathit{x}\right)=0$, the adaptation law for fuzzy system approximator is [6,28]:

$${\mathit{\phi }}_{\mathit{n}\mathit{e}\mathit{w}}^{\mathit{T}}={\mathit{\phi }}_{\mathit{o}\mathit{l}\mathit{d}}^{\mathit{T}}-{\mathsf{\Lambda }}^{-\mathbf{1}}{\mathsf{\sigma }}^{\mathit{T}}\mathit{z}\left(\mathit{x}\right).$$

(26)

Consequently, we obtain:

$$\dot{V}\le {-\mathsf{\sigma }}^{\mathit{T}}\mathit{M}{\mathit{w}}^{-\mathbf{1}}\mathsf{\sigma }\le 0,$$

(27)

which is negative definite. That means that the CDPR described in Equation (9), when controlled by a synergetic controller proposed in Equation (14) and combined with an adaptive fuzzy tracking controller, achieves global asymptotic stability in terms of $\mathsf{\sigma }$ and $\mathit{\phi }$ according to the Lyapunov test method.

Since synergetic control strategies are types of optimal controllers, they share the same industrial limitations in practical applications. One significant limitation is the increasing complexity, which grows with the complexity and nonlinearities of the plants. Additionally, their performance depends heavily on the accuracy of the plant model, making them sensitive to uncertainties and disturbances. The first limitation can be addressed by using digital controllers. The second can be mitigated by improving the controller design and incorporating adaptive methods to estimate and compensate for model changes.

3.4. DA Optimization

The classical DA, as introduced by [19], is a straightforward yet potent metaheuristic algorithm that emulates the swarm behavior of dragonflies. The social conduct of dragonflies, as observed during food search and collection, as well as enemy evasion, forms the foundation of the equation-based rules that drive DA. The primary DA equation is:

$$\Delta {\mathit{X}}_{k+1}=\left({\alpha }_1{\mathit{S}}_i+{\alpha }_2{\mathit{A}}_i+{\alpha }_3{\mathit{C}}_i+{\alpha }_4{\mathit{F}}_i+{\alpha }_5{\mathit{E}}_i\right)+\omega \Delta {\mathit{X}}_k$$

(28)

In this context, $\Delta {\mathit{X}}_{\mathit{k}}$ is the step vector of the position of the dragonfly at iteration k, and $\omega$ is the inertia weight. ${\mathit{S}}_i, {\mathit{A}}_i, {\mathit{C}}_i, {\mathit{F}}_i,$ and ${\mathit{E}}_i$ denote the separation, alignment, cohesion, food source, and position of the enemy of the $i^{th}$ individual, respectively. ${\alpha }_1, {\alpha }_2, {\alpha }_3, {\alpha }_4, \mathrm{a}\mathrm{n}\mathrm{d} {\alpha }_5$ represent the weights of each parameter, respectively. The parameters are calculated as follows:

$${\mathit{S}}_i={\sum }_{j=1}^N\left(\mathit{X}-{\mathit{X}}_j\right)$$

(29)

$${\mathit{A}}_i={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{\mathit{X}}_j$$

(30)

$${\mathit{C}}_i={\displaystyle \frac{1}{N}}\left({\sum }_{j=1}^N{\mathit{X}}_j\right)-\mathit{X}$$

(31)

$${\mathit{F}}_i={\mathit{X}}^{+}-\mathit{X}$$

(32)

$${\mathit{E}}_i={\mathit{X}}^{-}+\mathit{X}$$

(33)

where $\mathit{X}$ is the position of the current dragonfly, ${\mathit{X}}_j$ indicates the position of the $j^{th}$ neighboring solution, ${\mathit{X}}^{+}$ shows the position of a food source, ${\mathit{X}}^{-}$ is the position of an enemy, and N is the number of neighboring dragonflies.

If an adjacent solution exists, the position vector is calculated as:

$${\mathit{X}}_{k+1}=\left\{\begin{array}{cc}{\mathit{X}}_k+\Delta {\mathit{X}}_{k+1},\hfill & \mathrm{adjacent} \mathrm{solution} \mathrm{exists}\hfill \\ {\mathit{X}}_k+{\mathit{X}}_k\left(0.01\left({\displaystyle \frac{a_1}{{\left|a_2\right|}^{\frac{1}{b}}}}\right){\left({\displaystyle \frac{\left(q-1\right)!\left(1+b\right)\sin \left(\frac{b\pi }{2}\right)}{\left(q-1\right)!b.2^{\left(\frac{b-1}{2}\right)}}}\right)}^{{\displaystyle \frac{1}{b}}}\right),& \mathrm{otherwise}, \mathrm{explore} \mathrm{a} \mathrm{new} \mathrm{area}\hfill \end{array}\right.$$

(34)

where ${(0\le (a}_1$, $a_2)\le$ 1) are variables used to adjust explorations of new areas, b is a constant, and q is the dimension of $\mathit{X}$.

In swarm intelligence optimization, a balance between exploration (finding new search areas) and exploitation (intensive search within these areas) is essential. These stages manifest in DA through migration and feeding scenarios. Key swarm behaviors include separation (avoiding collisions with neighbor agents), alignment (matching the speed of neighboring agents), and cohesion (tendency to move toward the group center). Additionally, DA incorporates behaviors of moving toward food and avoiding enemies.

The main reasons behind choosing DA are that it is a straightforward algorithm, has fewer parameters to tune, and converges faster than other optimization algorithms in some fields like engineering [29,30]. It has its limitations also, like the tendency to prioritize exploitation over exploration which can lead to it getting stuck in local optima. It goes without saying that no single algorithm, including DA, can effectively solve all optimization problems [31,32].

Figure 3 illustrates the block diagram of the whole system and controller. DA is used to optimize the values of v and w of Equation (21).

4. Results and Discussion

The suspended CDPR system was modeled using MATLAB code. The kinematic and dynamic parameters of the CDPR were set as shown in

Table 2. Assigned values of kinematic and dynamic parameters of the suspended CDPR.

Parameter	Symbol	Value
Vector A1	[0, 0, d3]	[0, 0, 3] meters
Vector A2	[0, d2, d3]	[0, 4, 3] meters
Vector A3	[d1, 0, d3]	[4, 0, 3] meters
Vector A4	[d1, d2, d3]	[4, 4, 3] meters
Mass of EE	m	5 kg

, following [6]; that makes comparing the results fair. Reference [6] used the same model but with a different controller; therefore, it was employed to compare the results. It introduced a robust adaptive fuzzy tracking control strategy that consists of a sliding mode controller and an adaptive fuzzy logic system that approximates unknown functions and external disturbances, compensating for uncertainties in the model.

Figure 4 illustrates the simulation model developed using Simulink in MATLAB R2021a to validate the AFSC approach for a MIMO suspended CDPR. The adaptive fuzzy system is implemented to approximate nonlinear system dynamics, while the synergetic control framework ensures robust trajectory tracking and stability. Additionally, the model simulates environmental disturbances and parameter uncertainties to evaluate the controller’s performance under realistic operating conditions.

For demonstrating the effectiveness of the proposed controller, first, its performance was evaluated and optimized without any disturbance or uncertainty in the parameters. Then, the robustness was tested assuming a bounded disturbance and normal parameter uncertainty. DA is utilized to optimize the parameters of the controller. These parameters are the weights v and w of the synergetic controller. The objective function was selected to minimize the integral time absolute error (ITAE) performance index:

$$\mathrm{I}\mathrm{T}\mathrm{A}\mathrm{E}={\int }_0^{\infty }t\left|e\left(t\right)\right|dt.$$

(35)

Using this index leads to controllers that maintain the robustness of the system, minimize the overshoot in the response, and exhibit high load disturbance rejection [33].

The trajectory of the desired input to the system $P_d=[x_d y_{d }{z}_d]$ was chosen to be a circle, following [6], as follows:

$$x_d=0.8cos\left(0.1\pi t\right)+1.8, y_d=0.8sin\left(0.1\pi t\right)+2, z_d=1.5.$$

The initial value of the EE position $P_0=[x_0 y_{0 }{z}_0]$ was chosen to be on the path of the trajectory as [2.6, 2.0, 1.5]. This ensures a clear response, useful in comparison with [6]. The initial EE velocity ${\dot{P}}_0=[{\dot{x}}_0 {\dot{y}}_{0 }{\dot{z}}_0]$ was set to [0, 0, 0].

The radial basis fuzzy system was used as an adaptive compensation control for the composite disturbance $\mathit{\delta }$ of Equation (13). Its inputs are the measured positions and their derivates P and P which represent a Proportional–Derivative (PD) fuzzy system. The macro-variable of the manifold $\mathit{\sigma }$ of (10) was another input used in the adaptation law of (26). MFs were chosen to be five Gaussian functions for each input and five singletons for the output as shown in Figure 5. The choice of Gaussian MFs was to ensure a smooth approximation of the disturbance [34]. Singleton MFs were used for simplicity and fast calculations.

The standard deviation of each MF was set to 0.1, and the mean vector was set to [−0.6, −0.3, 0, 0.3, 0.6]. This ensures an even spread of MFs on the range [−1,1]. They represent five fuzzy values: negative large (NL), negative small (NS), zero (Z), positive small (PS), and positive large (PL). The rule base was set as shown in

Table 3. Rule base of the PD fuzzy system.

$\mathbf{e}/\dot{\mathit{e}}$	NL	NS	Z	PS	PL
NL	NL	NL	NL	NS	Z
NS	NL	NL	NS	Z	PS
Z	NL	NS	Z	PS	PL
PS	NS	Z	PS	PL	PL
PL	Z	PS	PL	PL	PL

[35]. This choice is common in controlling DC motors as in the case of a CDPR.

4.1. Scenario One: Without Disturbance and Parameter Uncertainities

For 250 iterations, DA, with a population size of 100 agents, was used to optimize w and v gains for the lowest value of ITAE. DA parameters were set as shown in

Table 4. Parameter settings of DA used for optimizing the synergetic controller gains.

Parameter	Symbol	Value (Case 1)
Number of variables to be optimized	s	6
Lower–upper bound of variables	lb, ub	0.1, 100
Nationhood hypersphere radius	r	$(ub-lb)(0.25+2{\displaystyle \frac{k}{ N}})$
Inertia weight	$\omega$	$(0.9+0.5{\displaystyle \frac{k}{ N}})$
Weight of separation, alignment, and cohesion	${\alpha }_1, {\alpha }_2, {\alpha }_3$	2Rand.$(0.1+0.2{\displaystyle \frac{k}{ N}})$
Weight of food factor	${\alpha }_4$	2Rand.
Weight of enemy factor	${\alpha }_5$	$(0.1+0.2{\displaystyle \frac{k}{ N}})$
Random walk parameters	[$a_1,a_2,b$]	[Rand., Rand., 1.5]

k is the iteration counter, N is the number of iterations.

following [36].

The optimized gains were found as follows:

$$\mathit{v}=\left[\begin{array}{ccc}12.4861& 0& 0\\ 0& 5.8431& 0\\ 0& 0& 11.6797\end{array}\right], \mathit{w}=\left[\begin{array}{ccc}14.6230& 0& 0\\ 0& 149.4794& 0\\ 0& 0& 200.00\end{array}\right],$$

with |ITAE| = 0.00308. The actual and desired positions of EE are shown in Figure 6.

4.2. Scenario Two: With Disturbance and Parameter Uncertainities

A simulation of disturbance and uncertainty signal was used to evaluate the robustness of AFSC. This signal is bounded and can be expressed as a vector [6], $\mathbf{D}={\left[4sin\left(10t\right), 2sin\left(10t\right), 4sin\left(10t\right)\right]}^T$. This signal was added to the τ vector. The tracking signal is illustrated in Figure 7.

Although ITAE is a good performance index in the optimization phase, it does not deliver the best depiction of error signal behavior. Two popular indices were used to show the error signal, which are the root-mean-square error (RMSE) and maximum absolute error (MAE). They are expressed as

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{t=1}^N}{\left|e\left(t\right)\right|}^2} \text{and} MAE=\underset{t=1\to N}{\max }\left|e\left(t\right)\right|.$$

(36)

Table 5. Error indices of the DA optimized AFSC and the robust adaptive fuzzy controller of [6].

System	RMSE	MAE
Robust adaptive fuzzy controller [6]	8.9867 × 10−4 m	2.0 × 10−2 m
DA Optimized AFSC	2.5307 × 10−6 m	7.2045 × 10−5 m

compares the system responses with the proposed DA optimized AFSC and the robust adaptive fuzzy controller proposed in [6].

Using a model that accounts for disturbances and uncertainties is essential because, in real-world applications, it’s nearly impossible to avoid them. Suspended CDPRs are especially vulnerable to disturbances due to their unique structure. Accurately modeling and compensating for these disturbances and uncertainties is therefore critical for ensuring robust and reliable operation in practical scenarios, particularly in industrial and outdoor applications where such influences are unavoidable.

5. Conclusions

This study presented AFSC optimized using DA for controlling suspended CDPRs. The proposed approach effectively addresses challenges related to external disturbances, and parameter uncertainties. By combining synergetic control theory’s robustness with the adaptive capabilities of the fuzzy basis system, the controller provides global asymptotic stability and accurate trajectory tracking. Simulation results under both ideal and disturbed conditions demonstrate the controller’s significant improvements in performance metrics, including lower tracking errors and enhanced robustness, compared to previously proposed controllers. These findings suggest that the DA optimized AFSC is a promising solution for improving the stability and control of suspended CDPRs in practical industrial applications. Future work could focus on experimental validation and extending this methodology to more complex multi-cable systems.