{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T08:06:13Z","timestamp":1784102773632,"version":"3.55.0"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T00:00:00Z","timestamp":1784073600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T00:00:00Z","timestamp":1784073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["None"],"award-info":[{"award-number":["None"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1007\/s00332-026-10267-7","type":"journal-article","created":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T07:22:25Z","timestamp":1784100145000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Strategies for Kiiking: Active Pumping to Invert a Swing"],"prefix":"10.1007","volume":"36","author":[{"given":"Petur","family":"Bryde","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ian C.","family":"Davenport","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L.","family":"Mahadevan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,15]]},"reference":[{"key":"10267_CR1","doi-asserted-by":"publisher","first-page":"1897","DOI":"10.1519\/JSC.0000000000001657","volume":"31","author":"HG Banyard","year":"2017","unstructured":"Banyard, H.G., Nosaka, K., Haff, G.G.: Reliability and validity of the load-velocity relationship to predict the 1RM back squat. J. Strength Condition. Res. 31, 1897 (2017)","journal-title":"J. Strength Condition. Res."},{"key":"10267_CR2","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6404\/aace82","volume":"39","author":"M Berry","year":"2018","unstructured":"Berry, M.: Pumping a swing revisited: minimal model for parametric resonance via matrix products. Eur. J. Phys. 39, 055007 (2018)","journal-title":"Eur. J. Phys."},{"key":"10267_CR3","doi-asserted-by":"publisher","unstructured":"Bryde, P., Davenport, I., Mahadevan, L.: Data and simulation code for \u201coptimal strategies for kiiking: active pumping to invert a swing\u201d. https:\/\/doi.org\/10.6084\/m9.figshare.22492993 (2023)","DOI":"10.6084\/m9.figshare.22492993"},{"key":"10267_CR4","doi-asserted-by":"publisher","first-page":"920","DOI":"10.1119\/1.1976498","volume":"38","author":"JA Burns","year":"1970","unstructured":"Burns, J.A.: More on pumping a swing. Am. J. Phys. 38, 920 (1970)","journal-title":"Am. J. Phys."},{"key":"10267_CR5","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1119\/1.18209","volume":"64","author":"WB Case","year":"1996","unstructured":"Case, W.B.: The pumping of a swing from the standing position. Am. J. Phys. 64, 215 (1996)","journal-title":"Am. J. Phys."},{"key":"10267_CR6","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1119\/1.16477","volume":"58","author":"WB Case","year":"1990","unstructured":"Case, W.B., Swanson, M.A.: The pumping of a swing from the seated position. Am. J. Phys. 58, 463 (1990)","journal-title":"Am. J. Phys."},{"key":"10267_CR7","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6404\/ab6a63","volume":"41","author":"P Glendinning","year":"2020","unstructured":"Glendinning, P.: Adaptive resonance and pumping a swing. Eur. J. Phys. 41, 025006 (2020)","journal-title":"Eur. J. Phys."},{"key":"10267_CR8","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1137\/1037043","volume":"37","author":"RF Hartl","year":"1995","unstructured":"Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37, 181 (1995)","journal-title":"SIAM Rev."},{"key":"10267_CR9","unstructured":"In addition, we estimate the center of mass of the athlete and measure the corresponding angle $$\\theta _{a}$$. As the athlete faces the same direction throughout, we have $$\\theta _{b} \\le \\theta \\le \\theta _{a}$$ where $$\\theta $$ is the angle of the equivalent simple pendulum. At certain times, when the athlete is standing up near the highest point of the swing, the angles $$\\theta _{a}$$ and $$\\theta _{b}$$ coincide (figure 1(b)). We estimate the $$\\theta $$ as the weighted sum $$\\theta = \\alpha \\theta _{a} + (1-\\alpha ) \\theta _{b}$$ where $$\\alpha $$ was chosen so that successive peaks of $$|{\\theta }|$$ increase monotonically as in figure 1(c). The difference $$|\\theta _{a} - \\theta _{b}|$$ serves as a bound on the error. The measured angle data is available in the supplementary material\u00a0[14]"},{"key":"10267_CR10","unstructured":"Kiiking: Estonian Kiiking Association, https:\/\/commons.wikimedia.org\/wiki\/File:Kiiking_07.JPG (2014). Accessed: 01-10-23"},{"key":"10267_CR11","doi-asserted-by":"crossref","unstructured":"Kulkarni, J.E.: Time-optimal control of a swing. In: 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475), vol.\u00a02, pp. 1729\u20131733. (2003) IEEE","DOI":"10.1109\/CDC.2003.1272862"},{"key":"10267_CR12","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/0021-8928(93)90059-U","volume":"57","author":"E Lavrovskii","year":"1993","unstructured":"Lavrovskii, E., Formal\u2019skii, A.: Optimal control of the pumping and damping of a swing. J. Appl. Math. Mech. 57, 311 (1993)","journal-title":"J. Appl. Math. Mech."},{"key":"10267_CR13","doi-asserted-by":"crossref","unstructured":"Luo, J., Lee, E.: Time-optimal control of the swing using impulse control actions. In: Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No. 98CH36207), vol.\u00a01, pp. 200\u2013204 (1998). IEEE","DOI":"10.1109\/ACC.1998.694658"},{"key":"10267_CR14","doi-asserted-by":"crossref","unstructured":"Maurer, H., Vossen, G.: Sufficient conditions and sensitivity analysis for optimal bang-bang control problems with state constraints. In: System Modeling and Optimization: 23rd IFIP TC 7 Conference, Cracow, Poland, 2007, Revised Selected Papers 23, pp. 82\u201399. Springer (2009)","DOI":"10.1007\/978-3-642-04802-9_4"},{"key":"10267_CR15","unstructured":"Note that this bound does not take into account the contribution $$u \\dot{u}$$ to $$\\dot{E}$$. This is done for the sake of simplicity; however, this term is usually small compared to the other two terms when the constraint is active, except at points where $$u$$ has a jump discontinuity"},{"key":"10267_CR16","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1109\/MCS.2005.1499390","volume":"25","author":"B Piccoli","year":"2005","unstructured":"Piccoli, B., Kulkarni, J.: Pumping a swing by standing and squatting: do children pump time optimally? IEEE Control Syst. Mag. 25, 48 (2005). https:\/\/doi.org\/10.1109\/MCS.2005.1499390","journal-title":"IEEE Control Syst. Mag."},{"key":"10267_CR17","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/PL00007956","volume":"84","author":"A Rahmani","year":"2001","unstructured":"Rahmani, A., Viale, F., Dalleau, G., Lacour, J.-R.: Force\/velocity and power\/velocity relationships in squat exercise. Eur. J. Appl. Physiol. 84, 227 (2001)","journal-title":"Eur. J. Appl. Physiol."},{"key":"10267_CR18","doi-asserted-by":"publisher","first-page":"676","DOI":"10.1038\/nmeth.2019","volume":"9","author":"J Schindelin","year":"2012","unstructured":"Schindelin, J., Arganda-Carreras, I., Frise, E., et al.: Fiji: an open-source platform for biological-image analysis. Nat. Methods 9, 676 (2012). https:\/\/doi.org\/10.1038\/nmeth.2019","journal-title":"Nat. Methods"},{"key":"10267_CR19","doi-asserted-by":"publisher","first-page":"1165","DOI":"10.1119\/1.1974385","volume":"36","author":"P Tea Jr","year":"1968","unstructured":"Tea, P., Jr., Falk, H.: Pumping on a swing. Am. J. Phys. 36, 1165 (1968)","journal-title":"Am. J. Phys."},{"key":"10267_CR20","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1080\/07468342.1998.11973953","volume":"29","author":"S Wirkus","year":"1998","unstructured":"Wirkus, S., Rand, R., Ruina, A.: How to pump a swing. Coll. Math. J. 29, 266 (1998)","journal-title":"Coll. Math. J."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-026-10267-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-026-10267-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-026-10267-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T07:22:29Z","timestamp":1784100149000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-026-10267-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,15]]},"references-count":20,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["10267"],"URL":"https:\/\/doi.org\/10.1007\/s00332-026-10267-7","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,15]]},"assertion":[{"value":"23 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 July 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"84"}}