{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:34:11Z","timestamp":1785544451561,"version":"3.56.0"},"reference-count":49,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T00:00:00Z","timestamp":1671148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012639","name":"Prince Sultan University","doi-asserted-by":"publisher","award":["Seed-CCIS-2022-117"],"award-info":[{"award-number":["Seed-CCIS-2022-117"]}],"id":[{"id":"10.13039\/501100012639","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this investigation, the adaptive fractional-order non-singular fixed-time terminal sliding mode (AFoFxNTSM) control for the uncertain dynamics of robotic manipulators with external disturbances is introduced. The idea of fractional-order non-singular fixed-time terminal sliding mode (FoFxNTSM) control is presented as the initial step. This approach, which combines the benefits of a fractional-order parameter with the advantages of NTSM, gives rapid fixed-time convergence, non-singularity, and chatter-free control inputs. After that, an adaptive control strategy is merged with the FoFxNTSM, and the resulting model is given the label AFoFxNTSM. This is done in order to account for the unknown dynamics of the system, which are caused by uncertainties and bounded external disturbances. The Lyapunov analysis reveals how stable the closed-loop system is over a fixed time. The pertinent simulation results are offered here for the purposes of evaluating and illustrating the performance of the suggested scheme applied on a PUMA 560 robot.<\/jats:p>","DOI":"10.3390\/e24121838","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T05:55:43Z","timestamp":1671429343000},"page":"1838","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Design of Adaptive Fractional-Order Fixed-Time Sliding Mode Control for Robotic Manipulators"],"prefix":"10.3390","volume":"24","author":[{"given":"Saim","family":"Ahmed","sequence":"first","affiliation":[{"name":"College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Automated Systems and Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7869-6373","authenticated-orcid":false,"given":"Ahmad Taher","family":"Azar","sequence":"additional","affiliation":[{"name":"College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Automated Systems and Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Faculty of Computers and Artificial Intelligence, Benha University, Benha 13518, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8883-6106","authenticated-orcid":false,"given":"Mohamed","family":"Tounsi","sequence":"additional","affiliation":[{"name":"College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Automated Systems and Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ahmed, S., Wang, H., and Tian, Y. (2016, January 27). Modification to model reference adaptive control of 5-link exoskeleton with gravity compensation. Proceedings of the Chinese Control Conference, Chengdu, China.","DOI":"10.1109\/ChiCC.2016.7554317"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.isatra.2016.09.009","article-title":"A hybrid robust fault tolerant control based on adaptive joint unscented Kalman filter","volume":"66","author":"Hagh","year":"2017","journal-title":"ISA Trans."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3198","DOI":"10.1109\/TMECH.2021.3055450","article-title":"Coordinated trajectory-tracking control of a marine aerial-surface heterogeneous system","volume":"26","author":"Wang","year":"2021","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1080\/00207170902769928","article-title":"A new terminal sliding mode control for robotic manipulators","volume":"82","author":"Zhao","year":"2009","journal-title":"Int. J. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2159","DOI":"10.1016\/S0005-1098(02)00147-4","article-title":"Non-singular terminal sliding mode control of rigid manipulators","volume":"38","author":"Feng","year":"2002","journal-title":"Automatica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1002\/rnc.1666","article-title":"Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems","volume":"21","author":"Yang","year":"2011","journal-title":"Int. J. Robust Nonlinear Control."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"110009","DOI":"10.1016\/j.automatica.2021.110009","article-title":"Fixed-time sliding mode control with mismatched disturbances","volume":"136","author":"Moulay","year":"2022","journal-title":"Automatica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ifacol.2018.07.079","article-title":"Continuous fixed-time sliding mode control for spacecraft with flexible appendages","volume":"51","author":"Ton","year":"2018","journal-title":"IFAC-PapersOnLine"},{"key":"ref_9","first-page":"1","article-title":"Applications of fractional operators in robotics: A review","volume":"104","year":"2022","journal-title":"J. Intell. Robot. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1007\/s12591-017-0362-x","article-title":"On Inverse Full State Hybrid Function Projective Synchronization for Continuous-time Chaotic Dynamical Systems with Arbitrary Dimensions","volume":"28","author":"Ouannas","year":"2020","journal-title":"Differ. Equ. Dyn. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ouannas, A., Azar, A.T., Ziar, T., and Radwan, A.G. (2017). Generalized Synchronization of Different Dimensional Integer-Order and Fractional Order Chaotic Systems, Springer.","DOI":"10.1007\/978-3-319-50249-6_23"},{"key":"ref_12","unstructured":"Azar, A.T., Radwan, A.G., and Vaidyanathan, S. (2018). Mathematical Techniques of Fractional Order Systems, Elsevier."},{"key":"ref_13","unstructured":"Azar, A.T., Radwan, A.G., and Vaidyanathan, S. (2018). Fractional Order Systems: Optimization, Control, Circuit Realizations and Applications, Elsevier."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Li, Y., Shang, Y., Duan, B., Cui, N., and Zhang, C. (2019). A fractional-Order kinetic battery model of lithium-Ion batteries considering a nonlinear capacity. Electronics, 8.","DOI":"10.3390\/electronics8040394"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Magin, R. (2004). Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng., 32.","DOI":"10.1615\/CritRevBiomedEng.v32.i1.10"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2020). Mathematical economics: Application of fractional calculus. Mathematics, 8.","DOI":"10.3390\/math8050660"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/B978-0-12-824293-3.00016-8","article-title":"Fractional calculus in electronic circuits: A review","volume":"1","author":"Tapadar","year":"2022","journal-title":"Fract. Order Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.isatra.2017.06.024","article-title":"Modeling and Analysis of Fractional Order DC-DC Converter","volume":"82","author":"Radwan","year":"2018","journal-title":"ISA Trans."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Meghni, B., Dib, D., Azar, A.T., Ghoudelbourk, S., and Saadoun, A. (2017). Robust Adaptive Supervisory Fractional order Controller For optimal Energy Management in Wind Turbine with Battery Storage, Springer.","DOI":"10.1007\/978-3-319-50249-6_6"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ouannas, A., Azar, A.T., Ziar, T., and Vaidyanathan, S. (2017). Fractional Inverse Generalized Chaos Synchronization Between Different Dimensional Systems, Springer.","DOI":"10.1007\/978-3-319-50249-6_18"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ouannas, A., Azar, A.T., Ziar, T., and Vaidyanathan, S. (2017). A New Method To Synchronize Fractional Chaotic Systems With Different Dimensions, Springer.","DOI":"10.1007\/978-3-319-50249-6_20"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2020\/3067024","article-title":"A Novel Design of a Neural Network based Fractional PID Controller for Mobile Robots Using Hybridized Fruit Fly and Particle Swarm Optimization","volume":"2020","author":"Ibraheem","year":"2020","journal-title":"Complexity"},{"key":"ref_23","first-page":"847","article-title":"TLBO Algorithm optimized fractional-order PID controller for AGC of interconnected power system","volume":"Volume 758","author":"Nayak","year":"2019","journal-title":"Soft Computing in Data Analytics. Advances in Intelligent Systems and Computing"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"43514","DOI":"10.1109\/ACCESS.2022.3167811","article-title":"Optimized Fractional Order Integral-Tilt Derivative Controller for Frequency Regulation of Interconnected Diverse Renewable Energy Resources","volume":"10","author":"Daraz","year":"2022","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"55","DOI":"10.24846\/v27i1y201806","article-title":"Fault tolerant control using fractional-order terminal sliding mode control for robotic manipulators","volume":"27","author":"Ahmed","year":"2018","journal-title":"Stud. Inform. Control."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Abro, G.E.M., Zulkifli, S.A.B., Asirvadam, V.S., and Ali, Z.A. (2021). Model-free-based single-dimension fuzzy SMC design for underactuated quadrotor UAV. Actuators, 10.","DOI":"10.3390\/act10080191"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ifacol.2018.06.014","article-title":"FOPID controllers and their industrial applications: A survey of recent results","volume":"51","author":"Tepljakov","year":"2018","journal-title":"IFAC-PapersOnLine"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3508","DOI":"10.1109\/TSMC.2021.3071360","article-title":"Novel neural network fractional-order sliding-mode control with application to active power filter","volume":"52","author":"Fei","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zheng, W., Luo, Y., Chen, Y., and Wang, X. (2021). A simplified fractional order PID controller\u2019s optimal tuning: A case study on a PMSM speed servo. Entropy, 23.","DOI":"10.3390\/e23020130"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.cnsns.2011.04.032","article-title":"Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty","volume":"17","author":"Dadras","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s40998-020-00364-y","article-title":"Output feedback adaptive fractional-order super-twisting sliding mode control of robotic manipulator","volume":"45","author":"Ahmed","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. Electr. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Fei, J., Wang, Z., and Liang, X. (2020). Robust adaptive fractional fast terminal sliding mode controller for microgyroscope. Complexity, 2020.","DOI":"10.1155\/2020\/8542961"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1007\/s11071-017-3570-6","article-title":"Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of fractional order chaotic systems","volume":"89","author":"Ni","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chen, D., Zhang, J., and Li, Z. (2022). A novel fixed-time trajectory tracking strategy of unmanned surface vessel based on the fractional sliding mode control method. Electronics, 11.","DOI":"10.3390\/electronics11050726"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Labbadi, M., Boubaker, S., Djemai, M., Mekni, S.K., and Bekrar, A. (2022). Fixed-Time Fractional-Order Global Sliding Mode Control for Nonholonomic Mobile Robot Systems under External Disturbances. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6040177"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2866","DOI":"10.1109\/TPWRS.2020.3043891","article-title":"Fixed-time fractional-order sliding mode controller for multimachine power systems","volume":"36","author":"Huang","year":"2020","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2737","DOI":"10.1016\/j.automatica.2014.10.015","article-title":"Multivariable adaptive control: A survey","volume":"50","author":"Tao","year":"2014","journal-title":"Automatica"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1454","DOI":"10.1109\/TCST.2015.2496585","article-title":"Adaptive robust finite-time trajectory tracking control of fully actuated marine surface vehicles","volume":"24","author":"Wang","year":"2015","journal-title":"IEEE Trans. Control. Syst. Technol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1322","DOI":"10.1109\/TSMC.2018.2834515","article-title":"Backpropagating constraints-based trajectory tracking control of a quadrotor with constrained actuator dynamics and complex unknowns","volume":"49","author":"Wang","year":"2018","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Lavretsky, E., and Wise, K.A. (2013). Robust Adaptive Control, Springer.","DOI":"10.1007\/978-1-4471-4396-3"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"729","DOI":"10.3390\/e16020729","article-title":"Robust control of a class of uncertain fractional-order chaotic systems with input nonlinearity via an adaptive sliding mode technique","volume":"16","author":"Tian","year":"2014","journal-title":"Entropy"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"76504","DOI":"10.1109\/ACCESS.2022.3192405","article-title":"Adaptive fractional-order non-singular fast terminal sliding mode control based on fixed time disturbance observer for manipulators","volume":"10","author":"Zhang","year":"2022","journal-title":"IEEE Access"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Lopes, A.M., and Machado, J.A.T. (2020). A review of fractional order entropies. Entropy, 22.","DOI":"10.3390\/e22121374"},{"key":"ref_44","first-page":"489","article-title":"Fast-exponential sliding mode control of robotic manipulator with super-twisting method","volume":"69","author":"Zhai","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/j.apm.2017.02.034","article-title":"Fractional-order exponential switching technique to enhance sliding mode control","volume":"44","author":"Yin","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1504\/IJMIC.2021.123799","article-title":"Robust model reference adaptive control for five-link robotic exoskeleton","volume":"39","author":"Ahmed","year":"2021","journal-title":"Int. J. Model. Identif. Control."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1991","DOI":"10.1049\/iet-cta.2016.0044","article-title":"Spacecraft fault-tolerant control using adaptive non-singular fast terminal sliding mode","volume":"10","author":"Han","year":"2010","journal-title":"IET Control Theory Appl."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Armstrong, B., Khatib, O., and Burdick, J. (1986, January 7\u201310). The explicit dynamic model and inertial parameters of the PUMA 560 arm. Proceedings of the 1986 IEEE International Conference on Robotics and Automation, San Francisco, CA, USA.","DOI":"10.1109\/ROBOT.1986.1087644"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1109\/TSMC.2019.2895588","article-title":"Adaptive high-order terminal sliding mode control based on time delay estimation for the robotic manipulators with backlash hysteresis","volume":"51","author":"Ahmed","year":"2019","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1838\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:08Z","timestamp":1760146988000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1838"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,16]]},"references-count":49,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["e24121838"],"URL":"https:\/\/doi.org\/10.3390\/e24121838","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,16]]}}}