{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T01:25:39Z","timestamp":1771377939534,"version":"3.50.1"},"reference-count":59,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,25]],"date-time":"2022-05-25T00:00:00Z","timestamp":1653436800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Consejer\u00eda de Econom\u00eda, Ciencia y Agenda Digital (Junta de Extremadura)","award":["GR21168"],"award-info":[{"award-number":["GR21168"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Ionic polymer-metal composites (IPMCs) are electrically driven materials that undergo bending deformations in the presence of relatively low external voltages, exhibiting a great potential as actuators in applications in soft robotics, microrobotics, and bioengineering, among others. This paper presents an artificial eukaryotic flagellum (AEF) swimming robot made up of IPMC segments for the study of planar wave generation for robot propulsion by single and distributed actuation, i.e., considering the first flagellum link as an actuator or all of them, respectively. The robot comprises three independent and electrically isolated actuators, manufactured over the same 10 mm long IPMC sheet. For control purposes, a dynamic model of the robot is firstly obtained through its frequency response, acquired by experimentally measuring the flagellum tip deflection thanks to an optical laser meter. In particular, two structures are considered for such a model, consisting of a non-integer order integrator in series with a resonant system of both non-integer and integer order. Secondly, the identified models are analyzed and it is concluded that the tip displacement of each actuator or any IPMC point is characterized by the same dynamics, which remains unchanged through the link with mere variations of the gain for low-frequency applications. Based on these results, a controller robust to gain variations is tuned to control link deflection regardless of link length and enabling the implementation of a distributed actuation with the same controller design. Finally, the deflection of each link is analyzed to determine whether an AEF swimming robot based on IPMC is capable of generating a planar wave motion by distributed actuation.<\/jats:p>","DOI":"10.3390\/a15060181","type":"journal-article","created":{"date-parts":[[2022,5,25]],"date-time":"2022-05-25T08:41:33Z","timestamp":1653468093000},"page":"181","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Modeling and Control of IPMC-Based Artificial Eukaryotic Flagellum Swimming Robot: Distributed Actuation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4966-1494","authenticated-orcid":false,"given":"Jos\u00e9 Emilio","family":"Traver","sequence":"first","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Avda. de Elvas s\/n, 06006 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9756-7641","authenticated-orcid":false,"given":"Cristina","family":"Nuevo-Gallardo","sequence":"additional","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Avda. de Elvas s\/n, 06006 Badajoz, Spain"}]},{"given":"Paloma","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Avda. de Elvas s\/n, 06006 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5542-348X","authenticated-orcid":false,"given":"In\u00e9s","family":"Tejado","sequence":"additional","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Avda. de Elvas s\/n, 06006 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5039-0738","authenticated-orcid":false,"given":"Blas M.","family":"Vinagre","sequence":"additional","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Avda. de Elvas s\/n, 06006 Badajoz, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1109\/JPROC.2014.2385105","article-title":"Biomedical Applications of Untethered Mobile Milli\/Microrobots","volume":"103","author":"Sitti","year":"2015","journal-title":"Proc. IEEE"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"eaam6431","DOI":"10.1126\/scirobotics.aam6431","article-title":"Micro\/nanorobots for biomedicine: Delivery, surgery, sensing, and detoxification","volume":"2","author":"Li","year":"2017","journal-title":"Sci. Robot."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"303","DOI":"10.4155\/tde-2017-0113","article-title":"Micro- and nano-motors: The new generation of drug carriers","volume":"9","author":"Xu","year":"2018","journal-title":"Ther. Deliv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2002203","DOI":"10.1002\/advs.202002203","article-title":"Medical micro\/nanorobots in precision medicine","volume":"7","author":"Soto","year":"2020","journal-title":"Adv. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2875","DOI":"10.1039\/D0EN00570C","article-title":"Environmental and health risks of nanorobots: An early review","volume":"7","author":"Arvidsson","year":"2020","journal-title":"Environ. Sci. Nano"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"8088","DOI":"10.1039\/D0CS00309C","article-title":"Medical micro\/nanorobots in complex media","volume":"49","author":"Wu","year":"2020","journal-title":"Chem. Soc. Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s10544-021-00546-3","article-title":"Biomimetic soft micro-swimmers: From actuation mechanisms to applications","volume":"23","author":"Fu","year":"2021","journal-title":"Biomed. Microdevices"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.surg.2020.10.040","article-title":"Milestones for autonomous in vivo microrobots in medical applications","volume":"169","author":"Sun","year":"2021","journal-title":"Surgery"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Jiang, J., Yang, Z., Ferreira, A., and Zhang, L. (2022). Control and autonomy of microrobots: Recent progress and perspective. Adv. Intell. Syst.","DOI":"10.1002\/aisy.202100279"},{"key":"ref_10","unstructured":"Shimoyama, I. (1995, January 5\u20139). Scaling in microrobots. Proceedings of the 1995 IEEE\/RSJ International Conference on Intelligent Robots and Systems. Human Robot Interaction and Cooperative Robots, Pittsburgh, PA, USA."},{"key":"ref_11","unstructured":"Trimmer, W., and Jebens, R. (1989, January 14\u201319). Actuators for micro robots. Proceedings of the 1989 IEEE International Conference on Robotics and Automation, Scottsdale, AZ, USA."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1088\/0143-0807\/22\/6\/305","article-title":"Scaling laws in the macro-, micro-and nanoworlds","volume":"22","author":"Wautelet","year":"2001","journal-title":"Eur. J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1109\/MRA.2007.380641","article-title":"Robotics in the small, part I: Microbotics","volume":"14","author":"Abbott","year":"2007","journal-title":"IEEE Robot. Autom. Mag."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"096601","DOI":"10.1088\/0034-4885\/72\/9\/096601","article-title":"The hydrodynamics of swimming microorganisms","volume":"72","author":"Lauga","year":"2009","journal-title":"Rep. Prog. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1561\/2300000023","article-title":"Micro-scale mobile robotics","volume":"2","author":"Diller","year":"2013","journal-title":"Found. Trends Robot."},{"key":"ref_16","unstructured":"Sitti, M. (2017). Mobile Microrobotics, MIT Press."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2002047","DOI":"10.1002\/adma.202002047","article-title":"Trends in micro-\/nanorobotics: Materials development, actuation, localization, and system integration for biomedical applications","volume":"33","author":"Wang","year":"2021","journal-title":"Adv. Mater."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"64001","DOI":"10.1209\/0295-5075\/86\/64001","article-title":"Life at high deborah number","volume":"86","author":"Lauga","year":"2009","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1088\/1367-2630\/9\/12\/437","article-title":"Swimming, pumping and gliding at low Reynolds numbers","volume":"9","author":"Raz","year":"2007","journal-title":"New J. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1109\/TRO.2013.2251211","article-title":"Geometric swimming at low and high Reynolds numbers","volume":"29","author":"Hatton","year":"2013","journal-title":"IEEE Trans. Robot."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.ijnonlinmec.2013.04.012","article-title":"Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers","volume":"56","author":"Alouges","year":"2013","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_22","first-page":"20160425","article-title":"Optimization and small-amplitude analysis of Purcell\u2019s three-link microswimmer model","volume":"472","author":"Wiezel","year":"2016","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kadam, S., Joshi, K., Gupta, N., Katdare, P., and Banavar, R. (October, January 27). Trajectory tracking using motion primitives for the Purcell\u2019s swimmer. Proceedings of the 2017 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Prague, Czech Republic.","DOI":"10.1109\/IROS.2017.8206159"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.ifacsc.2018.02.002","article-title":"Geometry of locomotion of the generalized Purcell\u2019s swimmer: Modelling, controllability and motion primitives","volume":"4","author":"Kadam","year":"2018","journal-title":"IFAC J. Syst. Control"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"K\u00f3sa, G., Jakab, P., Hata, N., J\u00f3lesz, F., Neubach, Z., Shoham, M., Zaaroor, M., and Sz\u00e9kely, G. (2008, January 19\u201322). Flagellar swimming for medical micro robots: Theory, experiments and application. Proceedings of the 2nd IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob 2008), Scottsdale, Arizona.","DOI":"10.1109\/BIOROB.2008.4762812"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1528","DOI":"10.1109\/TMECH.2016.2520567","article-title":"Piezoelectric beam for intrabody propulsion controlled by embedded sensing","volume":"21","author":"Abadi","year":"2016","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ollero, A., Sanfeliu, A., Montano, L., Lau, N., and Cardeira, C. (2018). Artificial Flagellum Microrobot. Design and Simulation in COMSOL. ROBOT 2017: Third Iberian Robotics Conference, Springer International Publishing.","DOI":"10.1007\/978-3-319-70833-1"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.ejcon.2020.08.004","article-title":"Performance study of propulsion of N-link artificial eukaryotic flagellum swimming microrobot within a fractional order approach: From simulations to hardware-in-the-loop experiments","volume":"58","author":"Traver","year":"2021","journal-title":"Eur. J. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"025013","DOI":"10.1088\/0964-1726\/23\/2\/025013","article-title":"A traveling wave piezoelectric beam robot","volume":"23","author":"Hariri","year":"2013","journal-title":"Smart Mater. Struct."},{"key":"ref_30","unstructured":"Prieto-Arranz, J., Traver, J.E., L\u00f3pez, M.A., Tejado, I., and Vinagre, B.M. (2018, January 5\u20137). Study in COMSOL of the generation of traveling waves in an AEF robot by piezoelectric actuation. Proceedings of the XXXIX Jornadas de Autom\u00e1tica, Badajoz, Espa\u00f1a."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"L\u00f3pez, M.A., Prieto, J., Traver, J.E., Tejado, I., Vinagre, B.M., and Petr\u00e1s, I. (2018, January 28\u201331). Testing non reciprocal motion of a swimming flexible small robot with single actuation. Proceedings of the 19th International Carpathian Control Conference (ICCC 2018), Szilv\u00e1sv\u00e1rad, Hungary.","DOI":"10.1109\/CarpathianCC.2018.8399647"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"eabf7364","DOI":"10.1126\/sciadv.abf7364","article-title":"Effect of body stiffness distribution on larval fish\u2013like efficient undulatory swimming","volume":"7","author":"Wang","year":"2021","journal-title":"Sci. Adv."},{"key":"ref_33","unstructured":"Dias, J.M.S. (2021). A Study on Bending Stiffness Characterization of Biohybrid Microrobots Using External Magnetic Actuation. [Ph.D. Thesis, Universidade de Coimbra]."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1088\/0964-1726\/14\/1\/020","article-title":"Ionic polymer metal composites: IV industrial and medical applications","volume":"14","author":"Shahinpoor","year":"2004","journal-title":"Smart Mater. Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"350","DOI":"10.5772\/56878","article-title":"Modelling and fuzzy control of an efficient swimming ionic polymer-metal composite actuated robot","volume":"10","author":"Shen","year":"2013","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1109\/JOE.2013.2259318","article-title":"Monolithic IPMC fins for propulsion and maneuvering in bioinspired underwater robotics","volume":"39","author":"Hubbard","year":"2013","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Chen, Z., Bart-Smith, H., and Tan, X. (2015). IPMC-actuated robotic fish. Robot Fish, Springer.","DOI":"10.1007\/978-3-662-46870-8_8"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40638-017-0081-3","article-title":"A review on robotic fish enabled by ionic polymer\u2013metal composite artificial muscles","volume":"4","author":"Chen","year":"2017","journal-title":"Robot. Biomim."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"071001","DOI":"10.1115\/1.4043101","article-title":"Robotic fish propelled by a servo motor and ionic polymer-metal composite hybrid tail","volume":"141","author":"Chen","year":"2019","journal-title":"J. Dyn. Syst. Meas. Control."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1063\/1.1495888","article-title":"Micromechanics of actuation of ionic polymer-metal composites","volume":"92","year":"2002","journal-title":"J. Appl. Phys."},{"key":"ref_41","unstructured":"Pugal, D. (2012). Physics Based Model of Ionic Polymer Metal Composite Electromechanical and Mechanoelectrical Transduction. [Ph.D. Thesis, University of Nevada]."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1109\/TMECH.2008.920021","article-title":"A control-oriented and physics-based model for ionic polymer\u2013metal composite actuators","volume":"13","author":"Chen","year":"2008","journal-title":"IEEE\/ASME Trans. Mechatronics"},{"key":"ref_43","first-page":"717659","article-title":"An enhanced fractional order model of ionic polymer-metal composites actuator","volume":"2013","author":"Caponetto","year":"2013","journal-title":"Adv. Math. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1146\/annurev.fl.09.010177.002011","article-title":"Fluid mechanics of propulsion by cilia and flagella","volume":"9","author":"Brennen","year":"1977","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_45","unstructured":"Happel, J., and Brenner, H. (2012). Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, Springer. [1st ed.]."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Hinch, E. (1988). Hydrodynamics at low Reynolds numbers: A brief and elementary introduction. Disorder and Mixing, Springer.","DOI":"10.1007\/978-94-009-2825-1_4"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1119\/1.10903","article-title":"Life at low Reynolds number","volume":"45","author":"Purcell","year":"1977","journal-title":"Am. J. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"R1","DOI":"10.1017\/jfm.2013.370","article-title":"Shape of optimal active flagella","volume":"730","author":"Lauga","year":"2013","journal-title":"J. Fluid Mech."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1242\/jeb.32.4.802","article-title":"The propulsion of sea-urchin spermatozoa","volume":"32","author":"Gray","year":"1955","journal-title":"J. Exp. Biol."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1017\/S0022112060001110","article-title":"Note on the swimming of slender fish","volume":"9","author":"Lighthill","year":"1960","journal-title":"J. Fluid Mech."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3321","DOI":"10.1063\/1.372343","article-title":"Electromechanical response of ionic polymer-metal composites","volume":"87","author":"Li","year":"2000","journal-title":"J. Appl. Phys."},{"key":"ref_52","unstructured":"Farinholt, K.M. (2005). Modeling and Characterization of Ionic Polymer Transducers for Sensing and Actuation. [Ph.D. Thesis, Virginia Tech]."},{"key":"ref_53","unstructured":"Caponetto, R., Graziani, S., Pappalardo, F.L., Umana, E., Xibilia, M., and Giamberardino, P.D. (2011, January 5\u20137). A scalable fractional model for IPMC actuator. Proceedings of the 7th Vienna International Conference on Mathematical Modelling, Vienna, Austria."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Caponetto, R., Dongola, G., Fortuna, L., Graziani, S., and Strazzeri, S. (2008, January 12\u201315). A fractional model for IPMC actuators. Proceedings of the 2008 IEEE Instrumentation and Measurement Technology Conference, Victoria, BC, Canada.","DOI":"10.1109\/IMTC.2008.4547395"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Tejado, I., Traver, J.E., Prieto-Arranz, J., L\u00f3pez, M.\u00c1., and Vinagre, B.M. (2019, January 25\u201328). Frequency domain based fractional order modeling of IPMC actuators for control. Proceedings of the 2019 18th European Control Conference (ECC), Naples, Italy.","DOI":"10.23919\/ECC.2019.8795739"},{"key":"ref_56","unstructured":"Val\u00e9rio, D., and S\u00e1 da Costa, J. (2004, January 19\u201321). Ninteger: A fractional control toolbox for Matlab. Proceedings of the 1st IFAC Workshop on Fractional Differentiation and its Applications, Bourdeaux, France."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s11071-004-3763-7","article-title":"Tuning of PID controllers based on Bode\u2019s ideal transfer function","volume":"38","author":"Barbosa","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., and Feliu-Batlle, V. (2010). Fractional-Order Systems and Controls: Fundamentals and Applications, Springer. [1st ed.].","DOI":"10.1007\/978-1-84996-335-0"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"12042","DOI":"10.3182\/20110828-6-IT-1002.02251","article-title":"Simple PID tuning rules with guaranteed Ms robustness achievement","volume":"44","author":"Arrieta","year":"2011","journal-title":"IFAC Proc. Vol."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/6\/181\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:18:11Z","timestamp":1760138291000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/6\/181"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,25]]},"references-count":59,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["a15060181"],"URL":"https:\/\/doi.org\/10.3390\/a15060181","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,25]]}}}