Abstract
In this article, an unknown system dynamics estimator-based impedance control method is proposed for the lower limb exoskeleton to stimulate the tracking flexibility with the terminal target position when suffering parametric inaccuracies and unexpected disturbances. To reinforce the robust performance, via constructing the filtering operation-based dynamic relation, i.e., invariant manifold, the unknown system dynamics estimators are employed to maintain the accurate perturbation identification in both the hip and knee subsystem. Besides, a funnel control technique is designed to govern the convergence process within a minor overshoot and a higher steady-state precision. Meanwhile, an interactive complaint result can be obtained with the aid of the impedance control, where the prescribed terminal trajectory can be adjusted into the interaction variable-based target position by the force–position mapping, revealing the dynamic influence between the impedance coefficient (stiffness and damping) and the adjusted position magnitude. A sufficient stability analysis verifies the ultimately uniformly bounded results of all the error signals, and even the angle errors can be regulated within the predefined funnel boundary in the whole convergence. Finally, some simulations are provided to demonstrate the validity and superiority including the enhanced interaction flexibility and robustness.
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References
Ibrahim, T., Shivesh, K., & Melya, B. (2022). A survey on design and control of lower extremity exoskeletons for bipedal walking. Applied Sciences-Basel, 12(5), 2395.
Arcos-Legarda, J., Torres, D., Velez, F., Rodriguez, H., Parra, A., & Gutierrez, A. (2023). Mechatronics design of a gait-assistance exoskeleton for therapy of children with Duchenne muscular dystrophy. Applied Sciences-Basel, 13(2), 839.
Masengo, G., Zhang, X. D., Dong, R. L., Alhassan, A. B., Hamza, K., & Mudaheranwa, E. (2023). Lower limb exoskeleton robot and its cooperative control: A review, trends, and challenges for future research. Frontiers in Neurorobotics, 16, 913748.
Susanto, S., Simorangkir, I. T., Analia, R., Analia, R., Pamungkas, D. S., Soebhakti, H., Sani, A., & Caesarendra, W. (2021). Real-time identification of knee joint walking gait as preliminary signal for developing lower limb exoskeleton. Electronics, 10(17), 2117.
Zhang, Y., Bressel, M., De Groof, S., Labey, L., & Peyrodie, L. (2023). Design and control of a size-adjustable pediatric lower-limb exoskeleton based on weight shift. IEEE Access, 11, 6372–6384.
Park, K. W., Choi, J., & Kong, K. (2023). Hybrid filtered disturbance observer for precise motion generation of a powered exoskeleton. IEEE Transactions on Industrial Electronics, 70(1), 646–656.
Sun, Z. X., Qiu, J. X., Zhu, J. H., & Li, S. (2022). A composite position control of flexible lower limb exoskeleton based on second-order sliding mode. Nonlinear Dynamics, 111(2), 1657–1666.
Dao, Q. T., Dinh, V., Trinh, M. C., Nguyen, V. L., Duong, M. D., & Bui, N. T. (2022). Nonlinear extended observer-based ADRC for a lower-limb PAM-based exoskeleton. Actuators, 11(12), 369.
Wang, J., Liu, J. H., Zhang, G. W., & Guo, S. (2022). Periodic event-triggered sliding mode control for lower limb exoskeleton based on human-robot cooperation. ISA Transactions, 123, 87–97.
Shao, X. L., Li, S. X., Zhang, W. D., & Wu, E. Q. (2023). Distance-based elliptical circumnavigation control for non-holonomic robots with event-triggered unknown system dynamics estimators. IEEE Transactions on Intelligent Transportation Systems, 24(4), 3986–3998.https://doi.org/10.1109/TITS.2023.3234369
Na, J., Yang, J., Wang, S. B., Gao, G., & Yang, C. (2021). Unknown dynamics estimator-based output-feedback control for nonlinear pure-feedback systems. IEEE Transactions on Systems Man Cybernetics-Systems, 51(6), 3832–3843.
Xing, Y. S., Na, J., Chen, M. R., Costa-Castello, R., & Roda, V. (2022). Adaptive nonlinear parameter estimation for a proton exchange membrane fuel cell. IEEE Transactions on Power Electronics, 37(8), 9012–9023.
Wang, S. B., Tao, L., Chen, Q., Na, J., & Ren, X. (2020). USDE-based sliding mode control for servo mechanisms with unknown system dynamics. IEEE-ASME Transactions on Mechatronics, 25(2), 1056–1066.
He, S. D., Dai, S. L., Zhao, Z. J., & Zou, T. (2022). Uncertainty and disturbance estimator-based distributed synchronization control for multiple marine surface vehicles with prescribed performance. Ocean Engineering, 261, 111867.
Ilchmann, A., Ryan, E. P., & Townsend, P. (2006). Tracking control with prescribed transient behaviour for systems of known relative degree. System and Control Letters, 55(5), 396–406.
Zahedi, M., & Binazadeh, T. (2020). Robust output tracking of nonlinear systems with transient improvement via funnel-based sliding mode control. Transactions of the Institute of Measurement and Control, 42(16), 3225–3233.
Wang, S. B., Li, S. Q., Chen, Q., Ren, X., & Yu, H. (2022). Funnel tracking control for nonlinear servo drive systems with unknown disturbances. ISA Transactions, 128, 328–335.
Lee, J. G., Trenns, S., & Shim, H. (2022). Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling. Automatica, 141, 110276.
Zhang, W. H., Shao, X. L., Zhang, W. D., Qi, J., & Li, H. (2022). Unknown input observer-based appointed-time funnel control for quadrotors. Aerospace Science and Technology, 126, 107351.
Sun, H. B., Hou, L. L., & Wei, Y. L. (2022). Decentralized dynamic event-triggered output feedback adaptive fixed-time funnel control for interconnection nonlinear systems. IEEE Transactions on Neural Network and Learning Systems. https://doi.org/10.1109/TNNLS.2022.3183290
Min, X., Baldi, S., & Yu, W. W. (2022). Distributed output feedback funnel control for uncertain nonlinear multiagent systems. IEEE Transactions on Fuzzy Systems, 30(9), 3708–3721.
Liu, L., Leonhardt, S., Bergmann, L., & Misgeld, B. J. (2022). Composite performance of variable stiffness actuator for exoskeleton administrated via impedance control and disturbance observer. Mechanism and Machine Theory, 179, 105096.
Li, Z. J., Li, Q. J., Huang, P. B., Xia, H., & Li, G. (2023). Human-in-the-loop adaptive control of a soft exo-suit with actuator dynamics and ankle impedance adaptation. IEEE Transactions on Cybernetics, 53(12), 7920–7932. https://doi.org/10.1109/TCYB.2023.3240231
Foroutannia, A., Akbarzadeh-T, M. R., Akbarzadeh, A., & Tahamipour-Z, S. M. (2023). Adaptive fuzzy impedance control of exoskeleton robots with electromyography-based convolutional neural networks for human intended trajectory estimation. Mechatronics, 91, 102952.
Xiong, X. F., Do, C. D., & Manoonpong, P. (2022). Learning-based multifunctional elbow exoskeleton control. IEEE Transactions on Industrial Electronics, 39(9), 9216–9224.
Yang, Y., Huang, D. Q., Jin, C. W., Liu, X., & Li, Y. (2022). Neural learning impedance control of lower limb rehabilitation exoskeleton with flexible joints in the presence of input constraints. International Journal of Robust and Nonlinear Control, 33(7), 4191–4209. https://doi.org/10.1002/rnc.6390
Gonzalez, J. A. C., Salas-Pena, O., & De Leon-Morales, J. (2021). Observer-based super twisting design: A comparative study on quadrotor altitude control. ISA Transactions, 109, 307–314.
Dalla Gasperina, S., Longatelli, V., Braghin, F., Pedrocchi, A., & Gandolla, M. (2022). Development and electromyographic validation of a compliant human-robot interaction controller for cooperative and personalized neurorehabilitation. Frontiers in Neurorobotics, 15, 734130.
Chen, Z. L., Guo, Q., Li, T. S., Yan, Y., & Jiang, D. (2022). Gait prediction and variable admittance control for lower limb exoskeleton with measurement delay and extended-state-observer. IEEE Transactions on Neural Networks and Learning Systems, 34(11), 8693–8706. https://doi.org/10.1109/TNNLS.2022.3152255
Wang, Y.F., Liu, Z., Zhu, L.C., Li, X., & Wang, H. (2020). An impedance control method of lower limb exoskeleton rehabilitation robot based on predicted forward dynamics. 2020 IEEE 19th International Conference on Trust, Security and Privacy in Computing and Communications, Guangzhou, China, pp. 1515–1518.
Shao, X. L., Zhang, J. T., Zhang, W. D., & Zuo, Z. (2023). Multi-circular formation control with reinforced transient profiles for nonholonomic vehicles: A path-following framework. Defence Technology. https://doi.org/10.1016/j.dt.2023.02.023
Zhang, F., Shao, X. L., Xia, Y., & Zhang, W. (2023). Elliptical encirclement control capable of reinforcing performances for UAVs around a dynamic target. Defence Technology. https://doi.org/10.1016/j.dt.2023.03.014
Rojas, H. D., & Cortes-Romero, J. (2023). On the equivalence between generalized proportional integral observer and disturbance observer. ISA Transactions, 133, 397–411.
Sun, J. K., & Zeng, Z. G. (2023). Disturbance rejection control for networked control systems using subpredictor-based extended state observer. IEEE Transactions on Systems Man Cybernetics Systems, 53(6), 3629–3639. https://doi.org/10.1109/TSMC.2022.3230045
Zhang, J. T., Shao, X. L., Zhang, W. D., & Na, J. (2023). Path-following control capable of reinforcing transient performances for networked mobile robots over a single curve. IEEE Transactions on Instrumentation and Measurement, 72, 3513–312. https://doi.org/10.1109/TIM.2022.3201930
Long, Y., & Peng, Y. J. (2022). Extended state observer-based nonlinear terminal sliding mode control with feedforward compensation for lower extremity exoskeleton. IEEE Access, 10, 8643–8652.
Lv, J. X., Wang, C. H., Kao, Y. G., et al. (2023). A fixed-time distributed extended state observer for uncertain second-order nonlinear system. ISA Transactions, 138, 373–383. https://doi.org/10.1016/j.isatra.2023.02.016
Shao, X. L., Li, S. X., Zhang, W. D., & Zhang, Q. (2023). GPS-free collaborative elliptical circumnavigation control for multiple non-holonomic vehicles, 8(6), 3750–3761. IEEE Transaction on Intelligent Vehicles. https://doi.org/10.1109/TIV.2023.3240855
Li, S. X., Shao, X. L., Zhang, W. D., & Zhang, Q. (2023). Distributed multicircular circumnavigation control for UAVs with desired angular spacing. Defence Technology. https://doi.org/10.1016/j.dt.2023.02.007
Rios, H., Falcon, R., Gonzales, O. A., & Dzul, A. (2019). Continuous sliding-mode control strategies for quadrotor robust tracking: Real-time application. IEEE Transactions on Industrial Electronics, 66(2), 1264–1272.
Chen, W. H., Yang, J., Guo, L., & Li, S. (2016). Disturbance-observer-based control and related methods-an overview. IEEE Transactions on Industrial Electronics, 63(2), 1083–1095.
Aole, S., Elamvazuthi, I., Waghmare, L., Patre, B., Bhaskarwar, T., Meriaudeau, F., & Su, S. (2022). Active disturbance rejection control based sinusoidal trajectory tracking for an upper limb robotic rehabilitation exoskeleton. Applied Science-Basel, 12(3), 1287.
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This work was supported in part by the Young Talent Fund of Association for Science and Technology in Shaanxi, China (No. 20230126).
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Zhang, W., Song, P., Wu, M. et al. Unknown system dynamics estimator-based impedance control for lower limb exoskeleton with enhanced performance. Control Theory Technol. 22, 56–68 (2024). https://doi.org/10.1007/s11768-023-00189-0
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DOI: https://doi.org/10.1007/s11768-023-00189-0