{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T13:27:27Z","timestamp":1773840447757,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"crossref","award":["PID2022-141409OB-C22\/AEI\/10.13039\/501100011033\/FEDER, UE"],"award-info":[{"award-number":["PID2022-141409OB-C22\/AEI\/10.13039\/501100011033\/FEDER, UE"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"crossref"}]},{"name":"European Regional Development Fund (FEDER) \u201cA way to make Europe\u201d"},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["FPU22\/00885"],"award-info":[{"award-number":["FPU22\/00885"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>Solenoid valves are widely used for pressure regulation in soft pneumatic robots, but their inherent electromechanical nonlinearities\u2014such as dead zones, saturation, and pressure-dependent dynamics\u2014pose significant challenges for accurate control. Conventional pulse modulation techniques, including pulse-width modulation (PWM), often exacerbate these effects by neglecting valve-switching transients. This paper presents a physics-informed dynamic modelling framework that captures transient and pressure-dependent behaviours in solenoid valve-driven soft pneumatic systems operating under pulse modulation. The model is experimentally validated on a soft pneumatic actuator (SPA) platform using four modulation schemes: PWM, integral pulse frequency modulation (IPFM), its inverted variant (IIPFM), and \u0394\u03a3 modulation. Results demonstrate that only the IIPFM scheme produces near-linear input\u2013pressure characteristics, in close agreement with model predictions. The proposed framework provides new physical insights into valve-induced nonlinearities and establishes a systematic basis for high-fidelity modelling and control of soft pneumatic robotic systems.<\/jats:p>","DOI":"10.3390\/robotics14120177","type":"journal-article","created":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T11:19:51Z","timestamp":1764328791000},"page":"177","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Open-Loop Characterisation of Soft Actuator Pressure Regulated by Pulse-Driven Solenoid Valve"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1385-7452","authenticated-orcid":false,"given":"Andr\u00e9s J.","family":"Serrano-Balbont\u00edn","sequence":"first","affiliation":[{"name":"Departamento de Ingenier\u00eda El\u00e9ctrica, Electr\u00f3nica y Autom\u00e1tica, Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, 06006 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5542-348X","authenticated-orcid":false,"given":"In\u00e9s","family":"Tejado","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda El\u00e9ctrica, Electr\u00f3nica y Autom\u00e1tica, Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, 06006 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5039-0738","authenticated-orcid":false,"given":"Blas M.","family":"Vinagre","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda El\u00e9ctrica, Electr\u00f3nica y Autom\u00e1tica, Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, 06006 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1691-1648","authenticated-orcid":false,"given":"Sumeet S.","family":"Aphale","sequence":"additional","affiliation":[{"name":"Artificial Intelligence, Robotics and Mechatronic Systems Group (ARMS), School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2394-3322","authenticated-orcid":false,"given":"Andres","family":"San-Millan","sequence":"additional","affiliation":[{"name":"Artificial Intelligence, Robotics and Mechatronic Systems Group (ARMS), School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hasanshahi, B., Cao, L., Song, K.Y., and Zhang, W. (2024). Design of Soft Robots: A Review of Methods and Future Opportunities for Research. Machines, 12.","DOI":"10.3390\/machines12080527"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2400414","DOI":"10.1002\/aisy.202400414","article-title":"A Review on Recent Trends of Bioinspired Soft Robotics: Actuators, Control Methods, Materials Selection, Sensors, Challenges, and Future Prospects","volume":"7","author":"Sarker","year":"2024","journal-title":"Adv. Intell. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wang, Y., Wang, Y., Mushtaq, R.T., and Wei, Q. (2024). Advancements in Soft Robotics: A Comprehensive Review on Actuation Methods, Materials, and Applications. Polymers, 16.","DOI":"10.3390\/polym16081087"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1146\/annurev-control-062322-100607","article-title":"An Overview of Soft Robotics","volume":"6","author":"Yasa","year":"2023","journal-title":"Annu. Rev. Control Robot. Auton. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6448","DOI":"10.1109\/LRA.2024.3406201","article-title":"Programmable Pressure Control in Pneumatic Soft Robots With 2-Way 2-State Solenoid Valves","volume":"9","author":"Chen","year":"2024","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"59442","DOI":"10.1109\/ACCESS.2022.3179589","article-title":"Soft Pneumatic Actuators: A Review of Design, Fabrication, Modeling, Sensing, Control and Applications","volume":"10","author":"Xavier","year":"2022","journal-title":"IEEE Access"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s11107-018-0815-3","article-title":"Optimal control scheme for pneumatic soft actuator under comparison of proportional and PWM-solenoid valves","volume":"37","author":"Huang","year":"2019","journal-title":"Photonic Netw. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"102797","DOI":"10.1016\/j.mechatronics.2022.102797","article-title":"Soft pneumatic actuators adapted in multiple environments: A novel fuzzy cascade strategy for the dynamics control with hysteresis compensation","volume":"84","author":"Liu","year":"2022","journal-title":"Mechatronics"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Visnevskis, K., Kassim, S.O., Elena Giannaccini, M., Vaziri, V., and Aphale, S.S. (2023, January 22\u201325). Improved Model of the PWM Driven 3\/2 Solenoid Valve Pneumatic System for Soft Pneumatic Actuators. Proceedings of the 2023 27th International Conference on Methods and Models in Automation and Robotics (MMAR), Mi\u0119dzyzdroje, Poland.","DOI":"10.1109\/MMAR58394.2023.10242591"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1919","DOI":"10.1109\/TMECH.2022.3155790","article-title":"Nonlinear Estimation and Control of Bending Soft Pneumatic Actuators Using Feedback Linearization and UKF","volume":"27","author":"Xavier","year":"2022","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/j.mechatronics.2004.09.007","article-title":"Hydraulic master\u2013slave land mine clearance robot hand controlled by pulse modulation","volume":"15","author":"Wojtara","year":"2005","journal-title":"Mechatronics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/s10015-024-00970-7","article-title":"Spiking neural networks-based generation of caterpillar-like soft robot crawling motions","volume":"29","author":"Yoshioka","year":"2024","journal-title":"Artif. Life Robot."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1177\/014233128000200202","article-title":"Design and implementation of pulse frequency modulation control systems","volume":"2","author":"Tzafestas","year":"1980","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_14","unstructured":"Serrano-Balbont\u00edn, A.J., Tejado, I., Vinagre, B.M., Aphale, S.S., and San-Millan, A. (IEEE Control Syst. Lett., 2025). Spiking Control of a Solenoid Valve for High-Precision Pressure Regulation in Soft Robotics, IEEE Control Syst. Lett., Under Review."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Norsworthy, S.R., Schreier, R., and Temes, G.C. (1996). Delta-Sigma Data Converters: Theory, Design, and Simulation, IEEE.","DOI":"10.1109\/9780470544358"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Aleixandre, M., Nakazawa, K., and Nakamoto, T. (2019). Optimization of Modulation Methods for Solenoid Valves to Realize an Odor Generation System. Sensors, 19.","DOI":"10.3390\/s19184009"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/S0925-4005(02)00300-3","article-title":"Study of odor blender using solenoid valves controlled by delta\u2013sigma modulation method for odor recorder","volume":"87","author":"Yamanaka","year":"2002","journal-title":"Sen. Actuators B Chem."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kassim, S.O., Visnevskis, K., Vaziri, V., and Aphale, S.S. (2023, January 20\u201322). A Dual Cavity Pleated Structures Soft Pneumatic Actuator for Soft Robotic Applications. Proceedings of the 2023 IEEE AFRICON, Nairobi, Kenya.","DOI":"10.1109\/AFRICON55910.2023.10293725"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Xavier, M.S., Fleming, A.J., and Yong, Y.K. (2022). Model-Based Nonlinear Feedback Controllers for Pressure Control of Soft Pneumatic Actuators Using On\/Off Valves. Front. Robot. AI, 9.","DOI":"10.3389\/frobt.2022.818187"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1115\/1.2897741","article-title":"Models of a Pneumatic PWM Solenoid Valve for Engineering Applications","volume":"114","author":"Ye","year":"1992","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Bolzern, P., and Spinelli, W. (July, January 30). Quadratic stabilization of a switched affine system about a nonequilibrium point. Proceedings of the 2004 American Control Conference, Boston, MA, USA.","DOI":"10.23919\/ACC.2004.1383918"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1109\/79.221324","article-title":"Signal processing with higher-order spectra","volume":"10","author":"Nikias","year":"1993","journal-title":"IEEE Signal Process. Mag."}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/14\/12\/177\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,30]],"date-time":"2025-11-30T05:26:41Z","timestamp":1764480401000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/14\/12\/177"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,28]]},"references-count":22,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["robotics14120177"],"URL":"https:\/\/doi.org\/10.3390\/robotics14120177","relation":{},"ISSN":["2218-6581"],"issn-type":[{"value":"2218-6581","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,28]]}}}