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Predefined-time fractional-order terminal SMC for robot dynamics
in increased responsiveness, lower tracking er-
ror, and greater control over nonlinear dynamics.
This work thoroughly addresses the constraints of
the proposed controller parameters and stability
proofs. The FoPtSMC approach’s proper param-
eters were carefully determined within designated
ranges to obtain the overall system’s stability and
error convergence within a fixed time. The study
underlines the need to select appropriate param-
eter values, such as a 1 , a 2 , a 3 > 0, b 1 , b 2 > 0,
0 < α, µ, η < 1, to achieve stability and conver-
Figure 4. Control input under uncertain dynamics.
gence within a specific time period; thereby sim-
plifying the process of picking acceptable values
The expected performance of the FoPtSMC
and increasing convergence speed.
method under uncertain dynamics in comparison
with 31 is effectively shown in Figures 2-4. Figure In conclusion, the suggested FoPtSMC
2 showcases the proposed controller’s precise abil- method has tremendous promise for increasing
ity to track the desired trajectory, while Figure 3 nonlinear systems’ tracking capabilities and ro-
exhibits minimal tracking errors of the proposed bustness, especially in the face of external distur-
scheme, affirming its effectiveness. Furthermore, bances. The work gives unique insights into the
Figure 4 illustrates the control inputs that guide development and application of the FoPtSMC ap-
the system state onto and maintain it on the de- proach, with implications for a wide range of con-
sired surfaces to ensure the desired dynamics, in- trol engineering applications.
dicating the proposed controller’s robust perfor-
mance in achieving accurate tracking. Addition-
ally, the proposed method delivers smooth con- 6. Conclusion
trol torques. The comprehensive assessment pre- The research focuses on controlling the trajectory
sented in Figures 2-4 emphasizes the efficacy of
tracking of robotic manipulators in the presence
the proposed FoPtSMC scheme in achieving pre-
of uncertainties and external disturbances. To
cise position trajectory tracking under uncertain
address this issue, a predefined-time fractional-
dynamics.
order sliding mode control (FoPtSMC) scheme is
5. Discussion proposed. This scheme allows the system state
to converge to the desired trajectory within a
The proposed FoPtSMC strategy has been de- predefined time, thereby improving tracking per-
signed to control nonlinear dynamical systems formance. Simulations on a manipulator with
with bounded external disturbances. This re- uncertain dynamics confirm the effectiveness of
search aims to improve tracking and transient FoPtSMC. It demonstrates faster response times,
characteristics while maintaining resilience in a reduced tracking errors, and improved rejection
closed-loop system. The FoPtSMC strategy is in- of uncertainties and disturbances. Moreover, fu-
tended to ensure rapid sliding mode convergence ture work should look into unknown dynamics in
within a specific time frame, and the Lyapunov nonlinear systems research, which will help ad-
technique was used to demonstrate stability. vance our understanding and application of the
The simulation findings show that the FoPtSMC technique.
FoPtSMC technique effectively manages a second-
order nonlinear dynamical robotic system under
external perturbations. The graphical depictions Acknowledgments
in Figures 2-3 display the variables x 1 , x 2 , x 3 ,
and the tracking error ε 1 , ε 2 , ε 3 , demonstrating This paper is derived from a research grant funded
the successful control performance and the abil- by the Research, Development, and Innovation
ity of the proposed system to reduce the track- Authority (RDIA), Kingdom of Saudi Arabia,
ing error to zero rapidly. Figure 4 depicts the with grant number 13382-psu-2023- PSNU-R-3-
control input, which has desirable qualities such 1-EI-. This research is supported by Automated
as smoothness and acceptable tracking perfor- Systems and Computing Lab (ASCL), Prince Sul-
mance, successfully mitigating the effects of ex- tan University, Riyadh, Saudi Arabia. The au-
ternal disturbances. The simulation adequately thors would like to thank Prince Sultan Univer-
illustrates and supports the theoretical analysis, sity, Riyadh, Saudi Arabia for supporting this
emphasizing the FoPtSMC approach’s advantages work.
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