{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T07:49:22Z","timestamp":1767772162417,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T00:00:00Z","timestamp":1724371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Faculty of Engineering of the University of Santiago of Chile and Agencia Nacional de Investigaci\u00f3n y Desarrollo de Chile","award":["2022-21220266"],"award-info":[{"award-number":["2022-21220266"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>This study proposes the design of a robust controller based on a Sliding Mode Control (SMC) structure. The proposed controller, called Sliding Mode Control based on Closed-Form Continuous-Time Neural Networks with Gravity Compensation (SMC-CfC-G), includes the development of an inverse model of the UR5 industrial robot, which is widely used in various fields. It also includes the development of a gravity vector using neural networks, which outperforms the gravity vector obtained through traditional robot modeling. To develop a gravity compensator, a feedforward Multi-Layer Perceptron (MLP) neural network was implemented. The use of Closed-Form Continuous-Time (CfC) neural networks for the development of a robot\u2019s inverse model was introduced, allowing efficient modeling of the robot. The behavior of the proposed controller was verified under load and torque disturbances at the end effector, demonstrating its robustness against disturbances and variations in operating conditions. The adaptability and ability of the proposed controller to maintain superior performance in dynamic industrial environments are highlighted, outperforming the classic SMC, Proportional-Integral-Derivative (PID), and Neural controllers. Consequently, a high-precision controller with a maximum error rate of approximately 1.57 mm was obtained, making it useful for applications requiring high accuracy.<\/jats:p>","DOI":"10.3390\/robotics13090126","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T12:53:19Z","timestamp":1724417599000},"page":"126","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Closed-Form Continuous-Time Neural Networks for Sliding Mode Control with Neural Gravity Compensation"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7197-8928","authenticated-orcid":false,"given":"Claudio","family":"Urrea","sequence":"first","affiliation":[{"name":"Electrical Engineering Department, Faculty of Engineering, University of Santiago of Chile, Las Sophoras 165, Estaci\u00f3n Central, Santiago 9170020, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7565-4859","authenticated-orcid":false,"given":"Yainet","family":"Garcia-Garcia","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, Faculty of Engineering, University of Santiago of Chile, Las Sophoras 165, Estaci\u00f3n Central, Santiago 9170020, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1958-7289","authenticated-orcid":false,"given":"John","family":"Kern","sequence":"additional","affiliation":[{"name":"Electrical Engineering Department, Faculty of Engineering, University of Santiago of Chile, Las Sophoras 165, Estaci\u00f3n Central, Santiago 9170020, Chile"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Urrea, C., and Garcia-Garcia, Y. (2023). Design and Performance Analysis of Level Control Strategies in a Nonlinear Spherical Tank. Processes, 11.","DOI":"10.3390\/pr11030720"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.robot.2018.08.008","article-title":"A supervisory on-line tuned fuzzy logic based sliding mode control for robotics: An application to surgical robots","volume":"109","author":"Qureshi","year":"2018","journal-title":"Robot. Auton. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sachan, S., and Swarnkar, P. (2023). Robust Motion Planning in Robot-Assisted Surgery for Nonlinear Incision Trajectory. Electronics, 12.","DOI":"10.3390\/electronics12030762"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1109\/TEC.2023.3247432","article-title":"Cascade control of grid-connected NPC converters via sliding mode technique","volume":"38","author":"Shen","year":"2023","journal-title":"IEEE Trans. Energy Convers."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"15406","DOI":"10.1109\/TPEL.2023.3313601","article-title":"Adaptive-gain second-order sliding mode control of NPC converters via super-twisting technique","volume":"38","author":"Shen","year":"2023","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"9189","DOI":"10.1109\/TPEL.2024.3386800","article-title":"Sliding Mode Control of Neutral-Point-Clamped Power Converters with Gain Adaptation","volume":"39","author":"Shen","year":"2024","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2588","DOI":"10.1002\/acs.3824","article-title":"Adaptive super twisting observer-based prescribed time integral sliding mode tracking control of uncertain robotic manipulators","volume":"38","author":"Shen","year":"2024","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Abbasi, S.J., and Lee, S. (2023). Enhanced Trajectory Tracking via Disturbance-Observer-Based Modified Sliding Mode Control. Appl. Sci., 13.","DOI":"10.3390\/app13148027"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Abadi, A., Ayeb, A., Labbadi, M., Fofi, D., Bakir, T., and Mekki, H. (2024). Robust Tracking Control of Wheeled Mobile Robot Based on Differential Flatness and Sliding Active Disturbance Rejection Control: Simulations and Experiments. Sensors, 24.","DOI":"10.3390\/s24092849"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"80138","DOI":"10.1109\/ACCESS.2024.3408829","article-title":"A Double Closed-Loop Digital Hydraulic Cylinder Position System Based on Global Fast Terminal Sliding Mode Active Disturbance Rejection Control","volume":"12","author":"Jiang","year":"2024","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chang, Y.-H., Yang, C.-Y., and Lin, H.-W. (2024). Robust Adaptive-Sliding-Mode Control for Teleoperation Systems with Time-Varying Delays and Uncertainties. Robotics, 13.","DOI":"10.3390\/robotics13060089"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1194","DOI":"10.1049\/cth2.12296","article-title":"Adaptive sliding-mode-assisted disturbance observer-based decoupling control for inertially stabilized platforms with a spherical mechanism","volume":"16","author":"Tian","year":"2022","journal-title":"IET Control. Theory Appl."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kern, J., Urrea, C., Verdejo, H., Agramonte, R., and Becker, C. (2024). Trajectory Tracking and Disturbance Rejection Performance Analysis of Classical and Advanced Controllers for a SCORBOT Robot. Robotics, 13.","DOI":"10.3390\/robotics13030048"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105836","DOI":"10.1016\/j.conengprac.2023.105836","article-title":"Sliding-mode control of a soft robot based on data-driven sparse identification","volume":"144","author":"Papageorgiou","year":"2024","journal-title":"Control. Eng. Pract."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.isatra.2023.11.013","article-title":"A model-free terminal sliding mode control for robots: Achieving fixed-time prescribed performance and convergence","volume":"144","author":"Truong","year":"2024","journal-title":"ISA Trans."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1002\/rcs.1784","article-title":"A fuzzy neural network sliding mode controller for vibration suppression in robotically assisted minimally invasive surgery","volume":"12","author":"Sang","year":"2016","journal-title":"Int. J. Med. Robotics Comput. Assist. Surg."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yuan, T., Zhang, C., Yi, F., Lv, P., Zhang, M., and Li, S. (2024). RBFNN-Based Adaptive Integral Sliding Mode Feedback and Feedforward Control for a Lower Limb Exoskeleton Robot. Electronics, 13.","DOI":"10.3390\/electronics13061043"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wu, H., Zhang, X., Song, L., Zhang, Y., Wang, C., Zhao, X., and Gu, L. (2023). Parallel Network-Based Sliding Mode Tracking Control for Robotic Manipulators with Uncertain Dynamics. Actuators, 12.","DOI":"10.3390\/act12050187"},{"key":"ref_19","first-page":"1","article-title":"Self-Organizing Type-2 Fuzzy Double Loop Recurrent Neural Network for Uncertain Nonlinear System Control","volume":"35","author":"Li","year":"2024","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1007\/s00202-023-02052-6","article-title":"Intelligent fractional-order sliding mode optimised control of surgical manipulator for healthcare system","volume":"106","author":"Sachan","year":"2024","journal-title":"Electr. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Khan, H., Khan, S.A., Lee, M.C., Ghafoor, U., Gillani, F., and Shah, U.H. (2023). DDPG-Based Adaptive Sliding Mode Control with Extended State Observer for Multibody Robot Systems. Robotics, 12.","DOI":"10.3390\/robotics12060161"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Li, T., Zhang, G., Zhang, T., and Pan, J. (2024). Adaptive Neural Network Tracking Control of Robotic Manipulators Based on Disturbance Observer. Processes, 12.","DOI":"10.3390\/pr12030499"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mystkowski, A., Wolniakowski, A., Kadri, N., Sewiolo, M., and Scalera, L. (2023). Neural Network Learning Algorithms for High-Precision Position Control and Drift Attenuation in Robotic Manipulators. Appl. Sci., 13.","DOI":"10.3390\/app131910854"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1038\/s42256-022-00556-7","article-title":"Closed-form continuous-time neural networks","volume":"4","author":"Hasani","year":"2022","journal-title":"Nat. Mach. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Hasani, R., Lechner, M., Amini, A., Rus, D., and Grosu, R. (2021, January 2\u20139). Liquid time-constant networks. Proceedings of the AAAI Conference on Artificial Intelligence, Virtual.","DOI":"10.1609\/aaai.v35i9.16936"},{"key":"ref_26","first-page":"1","article-title":"Neural ordinary differential equations","volume":"31","author":"Chen","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Urrea, C., Garcia-Garcia, Y., and Kern, J. (2024). Improving Surgical Scene Semantic Segmentation through a Deep Learning Architecture with Attention to Class Imbalance. Biomedicines, 12.","DOI":"10.3390\/biomedicines12061309"},{"key":"ref_28","unstructured":"Universal Robots A\/S (2024, June 16). UR5 Especificaciones T\u00e9cnicas No. Art\u00edculo 110105. Available online: https:\/\/www.universal-robots.com\/media\/50591\/ur5_es.pdf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1177\/1461348419874925","article-title":"An image vision and automatic calibration system for universal robots","volume":"40","author":"Jian","year":"2021","journal-title":"J. Low Freq. Noise Vib. Act. Control."},{"key":"ref_30","unstructured":"Kufieta, K. (2014). Force Estimation in Robotic Manipulators: Modeling, Simulation and Experiment. [Master\u2019s Thesis, NTNU Norwegian University of Science and Technology]. Available online: http:\/\/folk.ntnu.no\/tomgra\/Diplomer\/Kufieta.pdf."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Liu, J. (2017). Sliding Mode Control Using MATLAB, Academic Press.","DOI":"10.1016\/B978-0-12-802575-8.00005-9"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"138102","DOI":"10.1109\/ACCESS.2020.3012196","article-title":"Deep learning aided dynamic parameter identification of 6-DOF robot manipulators","volume":"8","author":"Wang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1007\/s40313-021-00687-z","article-title":"A robust Model free terminal sliding mode with gravity Compensation control of a 2 DoF exoskeleton-upper limb system","volume":"32","author":"Bembli","year":"2021","journal-title":"J. Control Autom. Electr. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Silaa, M.Y., Bencherif, A., and Barambones, O. (2024). Indirect Adaptive Control Using Neural Network and Discrete Extended Kalman Filter for Wheeled Mobile Robot. Actuators, 13.","DOI":"10.3390\/act13020051"}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/13\/9\/126\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:42:19Z","timestamp":1760110939000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/13\/9\/126"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,23]]},"references-count":34,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["robotics13090126"],"URL":"https:\/\/doi.org\/10.3390\/robotics13090126","relation":{},"ISSN":["2218-6581"],"issn-type":[{"type":"electronic","value":"2218-6581"}],"subject":[],"published":{"date-parts":[[2024,8,23]]}}}