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African vultures optimization-based hybrid neural network–proportional-integral-derivative controller...
controller yields the highest ITSE, reflecting the In all these cases, the NN–PID controller consis-
poorest performance. The figure clearly shows tently outperformed the other controllers. Specif-
that the actual trajectories of Psi-1, Psi-2, and ically, it achieved ITSE values of 0.28919 × 10 −4 ,
Psi-3 closely align with the desired trajectories 0.064321, 0.001164 across the respective tests,
when the NN–PID controller is used. whereas the STNN–PID controller yielded higher
Table 9 summarizes the performance charac- ITSE values of 3.54549×10 −4 , 3.526199, 0.883710,
teristics of all proposed controllers. Here, over- respectively. The most comprehensive and deci-
shoot is defined as the maximum deviation from sive evaluation involved a combined robustness
the desired trajectory at any point during the sim- test, incorporating all perturbations simultane-
ulation, regardless of disturbances. The data re- ously. The NN–PID controller demonstrated
veal strong competition between the con-PID and the highest resilience, achieving the lowest ITSE
NN–PID controllers. However, the NN–PID con- value of 0.073968, while the STNN–PID controller
troller consistently delivers slightly better perfor- recorded the weakest performance with an ITSE
mance. It exhibits the shortest rise time and set- value of 2.672754. These findings confirm that
tling time, with minimal overshoot, resulting in the NN–PID controller is the most effective con-
smooth and rapid convergence to the desired tra- troller in terms of tracking accuracy, stability, and
jectories. On the other hand, the results suggest robustness across all test scenarios. For future
that the NN–PID controller consumes more en- work, the proposed approach could be extended
ergy than the con-PID. by employing alternative metaheuristic optimiza-
Overall, across all tests and performance met- tion techniques—such as the dragonfly algorithm,
rics, the NN–PID controller outperforms the oth- artificial bee colony, cuckoo search, or salp swarm
ers, establishing itself as the most effective and algorithm—to optimize controller gains. Addi-
reliable control strategy among the proposed con- tionally, practical implementation and validation
trollers. using a physical robotic manipulator and the re-
quired hardware setup would strengthen the re-
sults. Comparative studies with other hybrid
7. Conclusions and future work schemes, such as the NN–PIPD controller, us-
ing the same manipulator model, are also recom-
Robotic manipulators are inherently complex,
mended to further evaluate the effectiveness of the
nonlinear, and highly coupled MIMO systems.
proposed strategy.
Their performance is significantly affected by ex-
ternal disturbances and uncertainties in system
parameters. As such, control strategies for these
Acknowledgments
systems must be capable of managing this com-
plexity while ensuring robustness and high track- This paper is derived from a research grant funded
ing accuracy. This research proposed two hy- by the Research, Development, and Innovation
brid controllers—STNN–PID and NN–PID—by Authority (RDIA), Kingdom of Saudi Arabia,
integrating the strengths of NNs and PID con- with grant number 13382-psu-2023-PSNU-R-3-1-
trollers to address the position tracking issue in a EI-. The authors would like to acknowledge
3-LRRM. These hybrid controllers were evaluated Prince Sultan University, Riyadh, Saudi Arabia,
against a con-PID controller. The parameters of for their support of this publication. This re-
all controllers were optimized using the AVOA, search is supported by the Automated Systems
aiming to minimize the ITSE as the performance and Computing Lab (ASCL), Prince Sultan Uni-
index. The ITSE was computed as the sum of versity, Riyadh, Saudi Arabia.
tracking errors for all three links. Under nom-
inal conditions, the NN–PID controller outper-
formed the others, yielding the lowest ITSE value Funding
of 0.31764 × 10 −4 , compared to 3.35957 × 10 −4
for the STNN–PID controller and 4.67050 × 10 −4 This work is funded by Prince Sultan University,
Riyadh, Saudi Arabia.
for the con-PID controller. The NN–PID con-
troller also demonstrated superior dynamic be-
havior, including minimal rise time, faster set-
Conflict of interest
tling time, and rapid convergence to the desired
trajectories. Robustness evaluations were con- The authors declare that the research was con-
ducted through individual tests involving varia- ducted in the absence of any commercial or fi-
tions in initial conditions, the introduction of ex- nancial relationships that could be construed as a
ternal disturbances, and parameter uncertainties. potential conflict of interest.
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