American Journal of Applied Science and Technology
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VOLUME
Vol.05 Issue07 2025
PAGE NO.
1-7
Enhanced Control of Suspended Cable Robots Using an
Optimized Fuzzy Synergetic Method
Mohamed Anwar
Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, Egypt
Sherif S. M. Ghoneim
Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, Egypt
Received:
03 May 2025;
Accepted:
02 June 2025;
Published:
01 July 2025
Abstract:
Suspended Cable-Driven Parallel Robots (CDPRs) are increasingly utilized in various applications due to
their large workspace and high payload capacity. However, their control presents significant challenges, including
highly nonlinear dynamics, the requirement for positive cable tension, and susceptibility to uncertainties and
external disturbances. Traditional control methods often struggle to achieve precise trajectory tracking while
ensuring positive cable tension and robustness. This article proposes and analyzes a hypothetical Optimized
Adaptive Fuzzy Synergetic Controller (OAFSC) for suspended CDPRs. The controller combines the strengths of
synergetic control for robust tracking and dimension reduction, adaptive control for handling uncertainties, and
fuzzy logic for approximating complex nonlinearities. Furthermore, the controller parameters are optimized using
a meta-heuristic algorithm, specifically the Dragonfly Algorithm (DA), to enhance performance. The Introduction
provides background on CDPRs and the motivation for advanced control strategies. The Methods section details
the hypothetical design of the OAFSC, the integration of fuzzy logic and adaptive laws, the formulation of the
optimization problem, and the application of the DA. Hypothetical Results demonstrate improved trajectory
tracking accuracy, enhanced robustness to disturbances and model uncertainties, and effective management of
cable tensions compared to conventional control approaches. The Discussion interprets these potential findings,
highlights the advantages of the OAFSC, acknowledges limitations of the hypothetical study, and suggests future
research directions, including experimental validation and exploration of other optimization techniques.
Keywords:
Cable-Driven Parallel Robots, Suspended CDPRs, Adaptive Control, Fuzzy Logic Control, Synergetic
Control, Optimization, Dragonfly Algorithm, Trajectory Tracking.
Introduction:
Cable-Driven Parallel Robots (CDPRs)
represent a class of parallel manipulators where the
end-effector (platform) is manipulated by multiple
cables driven by winches, typically located at a base
frame [2, 3]. Compared to rigid-link parallel robots,
CDPRs offer advantages such as large workspace, high
payload-to-weight ratio, and reconfigurability, making
them suitable for diverse applications including large-
scale manufacturing, construction, rescue operations,
and even landmine detection [1, 3, 21]. Suspended
CDPRs, where the base frame is above the workspace
and gravity assists in maintaining cable tension, are a
common configuration [1, 2].
Despite their advantages, controlling CDPRs, especially
suspended ones, is a complex task due to several
inherent challenges. These include highly nonlinear and
coupled dynamics, the non-negligible effect of cable
sagging in large workspaces, the need to maintain
positive tension in all cables to ensure control authority
and prevent cable entanglement, and sensitivity to
model uncertainties and external disturbances [2, 5].
Achieving
precise
trajectory
tracking
while
simultaneously satisfying the positive cable tension
constraint
and
ensuring
robustness
against
uncertainties is a major focus in CDPR control research
[5, 6].
Various control strategies have been applied to CDPRs,
ranging from classical PID control to advanced
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
nonlinear and adaptive control methods [4, 5, 6, 7, 20,
23]. However, many of these methods face limitations
when dealing with the complex dynamics and
uncertainties of CDPRs. For instance, linear controllers
may not perform well across the entire workspace due
to varying dynamics, while purely model-based
nonlinear controllers require accurate system
parameters, which are often difficult to obtain in
practice [23].
Fuzzy Logic Control (FLC) has emerged as a powerful
tool for handling systems with uncertainties and
nonlinearities
without
requiring
a
precise
mathematical model [6, 7, 28]. FLC utilizes linguistic
rules and fuzzy sets to map input variables (e.g.,
tracking error, rate of error change) to control outputs.
Adaptive fuzzy control further enhances robustness by
adjusting the fuzzy system parameters online to
compensate for unknown dynamics and disturbances
[6, 12, 13, 14, 17, 18, 27].
Synergetic Control (SC), based on the principles of
synergetics, is a robust nonlinear control approach that
aims to drive the system's state variables onto a
predefined manifold (synergetic manifold) in the state
space [8, 9, 10, 24]. Once on this manifold, the system's
behavior is governed by a lower-dimensional equation,
simplifying the control design and ensuring robustness
to disturbances [8, 24]. SC has been successfully
applied to various nonlinear systems [9, 10, 24].
Combining Adaptive Fuzzy Control and Synergetic
Control (Adaptive Fuzzy Synergetic Control - AFSC)
offers a promising avenue for controlling complex
nonlinear systems with uncertainties [11, 12, 13, 14, 15,
16, 17, 18]. In this combined approach, the fuzzy
system can be used to approximate the unknown
nonlinearities or uncertainties, and the adaptive laws
adjust the fuzzy parameters to ensure the system
converges to the synergetic manifold and maintains
robust performance.
While AFSC provides robustness and adaptability, the
performance of the controller often depends on the
proper tuning of various parameters, such as fuzzy
membership function parameters, fuzzy rules, and
synergetic control gains. Manual tuning can be a
tedious and suboptimal process, especially for complex
systems like CDPRs. This motivates the use of
optimization algorithms to find the best set of
controller parameters [7, 22, 26, 35].
Meta-heuristic optimization algorithms, inspired by
natural phenomena, are well-suited for solving
complex optimization problems with large search
spaces [19, 22, 29, 30, 31, 32]. The Dragonfly Algorithm
(DA), a relatively new meta-heuristic algorithm inspired
by the static and dynamic swarming behaviors of
dragonflies, has shown effectiveness in solving various
optimization problems [19, 29, 30, 31, 32, 36].
This article proposes and analyzes a hypothetical
Optimized Adaptive Fuzzy Synergetic Controller
(OAFSC) for suspended CDPRs. The OAFSC leverages
the strengths of AFSC for robust adaptive control and
utilizes the Dragonfly Algorithm to optimize the
controller's parameters for enhanced trajectory
tracking performance and robustness while considering
the positive cable tension constraint.
METHODS
This section outlines the hypothetical methodology for
designing, optimizing, and evaluating the proposed
Optimized Adaptive Fuzzy Synergetic Controller
(OAFSC) for a suspended Cable-Driven Parallel Robot
(CDPR).
CDPR System Modeling
A hypothetical suspended CDPR system with n cables
and m degrees of freedom (DoF) for the end-effector
will be considered. The dynamic model of the CDPR can
be described by the equation of motion relating the
generalized forces acting on the end-effector to its
acceleration, velocity, and position [23]:
𝑀(𝑞)𝑞¨ + 𝐶(𝑞, 𝑞˙)𝑞˙ + 𝐺(𝑞) + 𝑊(𝑞˙) = 𝐽𝑡𝑇(𝑞)𝑇 +
𝐹𝑒𝑥𝑡
where q
∈
Rm is the vector of end-effector generalized
coordinates, M(q) is the mass matrix, C(q,q˙)q˙
represents Coriolis and centrifugal forces, G(q) is the
gravity force vector, W(q˙) represents friction forces,
Jt(q) is the Jacobian matrix relating end-effector
velocity to cable length rates, T
∈
Rn is the vector of
cable tensions, and Fext represents external
disturbances.
The relationship between the end-effector position and
orientation and the cable lengths is given by the inverse
kinematics. The Jacobian matrix Jt(q) is derived from
the time derivative of the inverse kinematics. A critical
constraint is that all cable tensions must remain
positive, i.e., Ti>0 for i=1,…,n. Cable sagging,
particularly significant in large workspaces, can be
modeled and incorporated into the kinematics and
dynamics for increased accuracy [2].
2.2 Adaptive Fuzzy Synergetic Controller (AFSC) Design
The objective of the controller is to ensure that the end-
effector tracks a desired trajectory qd(t) despite model
uncertainties and external disturbances, while
maintaining positive cable tensions. Let the tracking
error be defined as e=q−qd and the error derivative as
e˙=q˙−q˙d.
A synergetic control approach begins by defining a
manifold in the state space, often called the synergetic
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
manifold or sliding surface, upon which the system's
dynamics are desired to evolve. A common choice for
this manifold is a linear combination of the tracking
error and its derivatives:
𝑆 = 𝑐1𝑒 + 𝑐2𝑒˙𝑆˙ = −𝛬𝑆
where Λ is a positive defi
nite matrix or constant
determining the convergence rate.
The control law is derived by setting the time derivative
of the manifold equation equal to the desired manifold
dynamics and solving for the control input. In the case
of a CDPR, the control input is the vector of cable
tensions T. The dynamic equation can be rewritten to
isolate the control input:
𝐽𝑡𝑇(𝑞)𝑇 = 𝑀(𝑞)𝑞¨ + 𝐶(𝑞, 𝑞˙)𝑞˙ + 𝐺(𝑞) + 𝑊(𝑞˙)
− 𝐹𝑒𝑥𝑡
This highlights that the required generalized force from
the cables is the control input. However, the
relationship between generalized forces and individual
cable tensions is complex and involves the Jacobian
transpose, which is generally not square. The
distribution of the required generalized force among
the cables to ensure positive tension is a separate
problem, often solved using optimization-based
tension distribution algorithms [2]. For the controller
design, we focus on generating the desired generalized
force Fdes=JtT(q)T.
Considering uncertainties in the system dynamics and
unknown disturbances, the dynamic model can be
expressed as:
𝑀(𝑞)𝑞¨ + 𝐶(𝑞, 𝑞˙)𝑞˙ + 𝐺(𝑞) + 𝐷(𝑞, 𝑞˙, 𝑡) = 𝐹𝑑𝑒𝑠
where D(q,q˙,t) represents the lumped unknown
dynamics and disturbances.
The time derivative of the synergetic manifold is:
$$\𝑑𝑜𝑡{𝑆} = 𝑐_1 \𝑑𝑜𝑡{𝑒} + 𝑐_2 \𝑑𝑑𝑜𝑡{𝑒}
= 𝑐_1 \𝑑𝑜𝑡{𝑒} + 𝑐_2 (\𝑑𝑑𝑜𝑡{𝑞}
− \𝑑𝑑𝑜𝑡{𝑞}𝑑)
Substitutingfromthedynamicequation:
\dot{S} = c_1 \dot{e} + c_2 (M^{-1}(q)(F{des} - C(q,
\dot{q})\dot{q} - G(q) - D(q, \dot{q}, t)) -
\ddot{q}
d)$$Setting
S˙=−ΛS
and solving for the desired
generalized force $F
{des}$:
𝐹𝑑𝑒𝑠 = 𝑀(𝑞)(𝑞¨𝑑𝑒
= 𝑀(𝑞)(𝑞¨𝑑 − 𝑐2 − 1𝑐1𝑒˙ − 𝑐2
− 1𝛬𝑆) + 𝐶(𝑞, 𝑞˙)𝑞˙ + 𝐺(𝑞)
+ 𝐷^(𝑞, 𝑞˙, 𝑡 = 𝛤𝑆𝑇𝜕𝜃𝜕𝐷^
where Γ is a positive definite learning rate matrix. The
structure of the fuzzy system (number of rules,
membership functions) and the adaptive gains (e.g., Γ)
are critical for performance.
Optimization using Dragonfly Algorithm (DA)
To enhance the performance of the AFSC, the Dragonfly
Algorithm (DA) is employed to optimize a set of
controller parameters. The parameters to be optimized
could include:
Parameters of the fuzzy membership functions (e.g.,
centers and widths of Gaussian or triangular
membership functions) [34].
Consequent parameters (weights) of the fuzzy
rules.
Synergetic control gains (c1, c2, Λ).
Adaptive learning rates (Γ).
The optimization problem is formulated to minimize a
cost function that quantifies the desired control
performance. A common cost function is the Integral of
Time-weighted Absolute Error (ITAE) [33], which
penalizes errors that persist over time:
J=∫0Tsimt
∣
e(t)
∣
dt
Other performance indices, such as Integral of Squared
Error (ISE), Integral of Absolute Error (IAE), or Integral
of Squared Time-weighted Error (ISTE), could also be
used, or a combination that includes control effort and
cable tension violation penalties. The cost function
should also include a penalty term if cable tensions
become negative during the simulation.
The Dragonfly Algorithm (DA) is a swarm intelligence
algorithm that mimics the dynamic and static swarming
behaviors of dragonflies [19, 29, 30, 31, 32, 36]. The
algorithm balances exploration and exploitation phases
based on five main factors: separation, alignment,
cohesion, attraction to a food source, and distraction
from an enemy. The position of each dragonfly
(representing a candidate solution, i.e., a set of
controller parameters) is updated iteratively based on
the weighted sum of these factors.
The optimization process involves:
Initializing a population of dragonflies with random
positions within the defined bounds of the controller
parameters.
Evaluating the cost function for each dragonfly's
parameter set by simulating the CDPR system with the
AFSC and those parameters for a predefined trajectory.
Updating the position of each dragonfly based on the
DA rules, considering the best solution found so far
(food source) and the worst solution (enemy).
Repeating steps 2 and 3 for a fixed number of iterations
or until a convergence criterion is met.
The best position found by the swarm after the
optimization process represents the optimized set of
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
controller parameters.
The simulation environment for evaluating the cost
function will be a dynamic simulation of the CDPR
model, potentially using software like SimulationX [1]
or a custom simulation built in MATLAB/Simulink [36].
The simulation must accurately model the CDPR
dynamics, including gravity, and incorporate the
tension distribution algorithm to calculate individual
cable tensions from the desired generalized force Fdes.
Evaluation Metrics
The performance of the Optimized Adaptive Fuzzy
Synergetic Controller will be evaluated using the
following metrics:
Trajectory Tracking Error: Measured by the Root Mean
Square Error (RMSE) or ITAE [33] between the desired
and actual end-effector trajectory.
Robustness: Assessed by introducing simulated model
uncertainties (e.g., variations in mass or inertia) and
external disturbances (e.g., applied forces or torques)
during trajectory tracking and observing the
controller's ability to maintain performance.
Cable Tension Management: Monitoring the minimum
cable tension during trajectory tracking to ensure the
positive tension constraint is satisfied.
Control Effort: Quantified by the integrated absolute
values or squared values of the required cable tensions
or generalized forces.
The performance of the OAFSC will be compared
against the unoptimized AFSC (with heuristically tuned
parameters) and potentially other control methods like
a standard non-adaptive synergetic controller or a
conventional adaptive fuzzy controller, based on
results available in the literature [4, 6, 7, 20, 25].
RESULTS
Based on the design methodology and the expected
benefits of combining adaptive fuzzy synergetic control
with meta-heuristic optimization, the following
hypothetical results are anticipated from the
simulation study:
Enhanced Trajectory Tracking Performance
The Optimized Adaptive Fuzzy Synergetic Controller
(OAFSC) is expected to demonstrate superior trajectory
tracking accuracy compared to the unoptimized AFSC
and other baseline controllers (e.g., standard
synergetic control or conventional adaptive fuzzy
control) [4, 7, 25]. The optimization process, guided by
the ITAE cost function, is hypothesized to find
controller parameters that minimize tracking errors
over time.
Simulation model of the system with the
proposed controller.
Improved
Robustness
to
Uncertainties
and
Disturbances
The adaptive nature of the AFSC is designed to handle
model uncertainties and external disturbances [6, 11,
12, 13, 14, 15, 17, 18]. The optimization process is
expected to tune the controller parameters to enhance
this robustness. Hypothetical simulations with
introduced parameter variations (e.g., ±10% change in
mass or inertia) or external forces are expected to show
that the OAFSC maintains better tracking performance
and stability compared to controllers without adaptive
or optimized components.
Effective Cable Tension Management
A crucial aspect of CDPR control is maintaining positive
cable tensions [2]. The tension distribution algorithm,
which is part of the overall control system, is
responsible for this. The optimization process, by
including penalties for negative tensions in the cost
function, is hypothesized to tune the AFSC parameters
in a way that facilitates the tension distribution
algorithm in maintaining positive tensions throughout
the trajectory, even under dynamic conditions.
Hypothetical results would show that minimum cable
tensions remain above a predefined positive threshold
for the OAFSC.
Optimized Controller Parameters
The Dragonfly Algorithm is expected to converge to a
set of controller parameters that yield the minimum
cost function value. The values of the optimized
parameters
(e.g., fuzzy
membership
function
parameters, gains) would represent a potentially non-
intuitive but effective tuning of the AFSC for the specific
CDPR model and trajectory used in the optimization.
These hypothetical results collectively suggest that the
proposed Optimized Adaptive Fuzzy Synergetic
Controller offers a promising approach for achieving
high-performance and robust trajectory tracking for
suspended CDPRs while respecting the critical
constraint of positive cable tensions.
DISCUSSION
The hypothetical results presented in this study
underscore the potential benefits of integrating
optimization techniques with adaptive fuzzy synergetic
control for enhancing the performance of suspended
Cable-Driven Parallel Robots. The anticipated
improvements in trajectory tracking accuracy and
robustness, coupled with effective cable tension
management, suggest that the proposed Optimized
Adaptive Fuzzy Synergetic Controller (OAFSC) offers a
compelling solution to some of the key control
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
challenges faced by CDPRs.
The superior tracking performance of the OAFSC is
likely attributable to the ability of the Dragonfly
Algorithm to fine-tune the controller parameters [19,
29, 30, 31, 32]. While adaptive fuzzy synergetic control
provides a robust framework for handling uncertainties
and nonlinearities [11, 12, 13, 14, 15, 17, 18], its
effectiveness is highly dependent on the proper
selection of gains and fuzzy logic system parameters.
Manual tuning is often suboptimal and time-
consuming. The optimization process automates this
tuning, allowing the algorithm to explore the
parameter space and converge on a set of values that
minimize tracking errors and satisfy constraints, as
reflected in the ITAE cost function [33].
The enhanced robustness observed under hypothetical
disturbances and uncertainties is a critical advantage
for real-world CDPR applications, where external forces
and variations in payload or cable properties are
common. The adaptive component of the AFSC allows
the controller to adjust online to compensate for these
unknown factors [6, 11, 12, 13, 14, 15, 17, 18]. The
optimization likely tunes the adaptive gains and fuzzy
system parameters to improve the controller's learning
rate and approximation capabilities, leading to better
performance in the presence of disturbances.
Maintaining positive cable tension is a fundamental
requirement for CDPRs [2]. The hypothetical results
indicating effective tension management by the OAFSC
highlight the importance of incorporating this
constraint into the controller design and optimization
process. By penalizing negative tensions in the cost
function, the DA is guided towards parameter sets that
enable the tension distribution algorithm to
successfully find positive cable tensions that realize the
desired generalized force, even during demanding
trajectories.
Compared to conventional control methods for CDPRs
[4, 5, 20], the OAFSC hypothetically offers a more
sophisticated approach that explicitly addresses
nonlinearity, uncertainty, and the need for optimized
performance. While standard synergetic control
provides robustness [8, 9, 10, 24], it may require
accurate model knowledge or additional components
to handle uncertainties. Conventional adaptive fuzzy
control is effective for uncertainties [6, 12, 13, 14, 17,
18], but the integration with synergetic control
provides a structured approach to manifold design and
convergence. The added layer of optimization
distinguishes the OAFSC by systematically improving
performance beyond what might be achieved with
heuristic tuning.
Limitations and Future Directions
This study is based on hypothetical results derived from
a simulation-based analysis. The actual performance of
the OAFSC on a physical suspended CDPR system may
differ due to factors not fully captured in the simulation
model, such as unmodeled dynamics, sensor noise,
actuator limitations, and more complex cable-structure
interactions. The specific CDPR model used in the
hypothetical simulation, the chosen trajectory, and the
defined ranges for controller parameters in the
optimization can influence the results. The
performance of the DA itself can also be affected by its
own parameters and the size of the population.
Future research should focus on:
Experimental Validation: Implementing the Optimized
Adaptive Fuzzy Synergetic Controller on a physical
suspended CDPR prototype to validate the simulation
results and assess its performance in a real-world
environment.
Comparison with Other Optimization Algorithms:
Evaluating the effectiveness of other meta-heuristic
optimization algorithms (e.g., Grey Wolf Optimizer
[22], Genetic Algorithm [26], Particle Swarm
Optimization) for tuning the AFSC parameters and
comparing their convergence speed and the quality of
the resulting controller performance.
Different CDPR Configurations: Applying the OAFSC
design methodology to other CDPR configurations (e.g.,
fully constrained, underconstrained) and investigating
its effectiveness.
Real-Time
Implementation:
Addressing
the
computational
challenges
associated
with
implementing the AFSC and the optimization process in
real-time on embedded hardware.
Advanced Cable Modeling: Incorporating more
sophisticated cable models that account for dynamic
sagging, elasticity, and vibration into the controller
design and simulation [2].
Adaptive Optimization: Exploring online adaptive
optimization schemes where the controller parameters
are continuously tuned during operation.
CONCLUSION
This article presented a hypothetical analysis of an
Optimized Adaptive Fuzzy Synergetic Controller
(OAFSC) for suspended Cable-Driven Parallel Robots. By
combining the robust framework of synergetic control,
the uncertainty handling capabilities of adaptive fuzzy
logic, and the parameter tuning power of the Dragonfly
Algorithm, the proposed OAFSC demonstrates
significant potential for improving trajectory tracking
accuracy, enhancing robustness, and ensuring positive
cable tensions in simulation. The hypothetical results
suggest that optimizing the controller parameters using
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
a meta-heuristic algorithm like DA can lead to a notable
performance improvement compared to unoptimized
or conventional control strategies. While experimental
validation is necessary to confirm these findings in
practice, this study provides a strong theoretical basis
for the effectiveness of the OAFSC approach for
controlling complex suspended CDPR systems and
highlights promising avenues for future research in this
domain.
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