{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T18:53:07Z","timestamp":1774637587426,"version":"3.50.1"},"reference-count":84,"publisher":"Association for Computing Machinery (ACM)","issue":"4","funder":[{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"crossref","award":["200021-231293"],"award-info":[{"award-number":["200021-231293"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2025,8,1]]},"abstract":"<jats:p>Numerous tasks in robotics and character animation involve solving combinations of inverse kinematics and motion planning problems that require the precise design of pose sequences to achieve desired motion objectives. Accounting for the complex interplay between body deformations and resulting motion, especially through interactions with the environment, poses significant challenges for the design of such pose sequences. We propose a computational framework to address these challenges in scenarios where the motion of a deformable body is entirely determined by dynamic changes of its shape. Complementing recent methods on the forward problem\u2014mapping shape sequences to global motion trajectories based on a geometric formulation of locomotion\u2014we address the inverse problem of optimizing shape sequences to achieve user-defined motion objectives. We demonstrate the effectiveness of our method through a diverse set of examples, producing realistic shape sequences that result in desired motion trajectories.<\/jats:p>","DOI":"10.1145\/3731187","type":"journal-article","created":{"date-parts":[[2025,7,27]],"date-time":"2025-07-27T04:02:22Z","timestamp":1753588942000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Inverse Geometric Locomotion"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0952-7063","authenticated-orcid":false,"given":"Quentin","family":"Becker","sequence":"first","affiliation":[{"name":"EPFL, Lausanne, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7970-5596","authenticated-orcid":false,"given":"Oliver","family":"Gross","sequence":"additional","affiliation":[{"name":"University of California, San Diego, San Diego, USA"},{"name":"EPFL, Lausanne, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4957-4825","authenticated-orcid":false,"given":"Mark","family":"Pauly","sequence":"additional","affiliation":[{"name":"EPFL, Lausanne, Switzerland"}]}],"member":"320","published-online":{"date-parts":[[2025,7,27]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/344779.344859"},{"key":"e_1_2_2_2_1","first-page":"6","article-title":"2018","volume":"37","author":"Aristidou A.","year":"2018","unstructured":"A. Aristidou, J. Lasenby, Y. Chrysanthou, and A. Shamir. 2018. Inverse Kinematics Techniques in Computer Graphics: A Survey. Comp. Graph. Forum 37, 6 (2018), 35\u201358.","journal-title":"Inverse Kinematics Techniques in Computer Graphics: A Survey. Comp. Graph. Forum"},{"key":"e_1_2_2_3_1","volume-title":"ACM SIGGRAPH 2009 Papers","author":"Barbi\u010d J.","unstructured":"J. Barbi\u010d, M. da Silva, and J. Popovi\u0107. 2009. Deformable object animation using reduced optimal control. In ACM SIGGRAPH 2009 Papers (New Orleans, Louisiana) (SIGGRAPH '09). Association for Computing Machinery, Article 53, 9 pages."},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s12220-024-01652-3"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112003005184"},{"key":"e_1_2_2_6_1","volume-title":"Actuators A La Mode: Modal Actuations for Soft Body Locomotion. In SIGGRAPH Asia 2024 Conference Papers. Article 53","author":"Benchekroun O.","unstructured":"O. Benchekroun, K. Xie, Hsueh-Ti D. Liu, E. Grinspun, S. Andrews, and V. Zordan. 2024. Actuators A La Mode: Modal Actuations for Soft Body Locomotion. In SIGGRAPH Asia 2024 Conference Papers. Article 53, 10 pages."},{"key":"e_1_2_2_7_1","volume-title":"Real: Learning Energy-Efficient Slithering Gaits for a Snake-Like Robot. 29, 4","author":"Bing Z.","year":"2022","unstructured":"Z. Bing, L. Cheng, K. Huang, and A. Knoll. 2022. Simulation to Real: Learning Energy-Efficient Slithering Gaits for a Snake-Like Robot. 29, 4 (2022), 92\u2013103."},{"key":"e_1_2_2_8_1","first-page":"1","article-title":"2023","volume":"39","author":"Bordalba R.","year":"2023","unstructured":"R. Bordalba, T. Schoels, L. Ros, J. M. Porta, and M. Diehl. 2023. Direct Collocation Methods for Trajectory Optimization in Constrained Robotic Systems. 39, 1 (2023), 183\u2013202.","journal-title":"Direct Collocation Methods for Trajectory Optimization in Constrained Robotic Systems."},{"key":"e_1_2_2_9_1","volume-title":"ACM SIGGRAPH 2004 Papers","author":"Botsch M.","unstructured":"M. Botsch and L. Kobbelt. 2004. An intuitive framework for real-time freeform modeling. In ACM SIGGRAPH 2004 Papers (Los Angeles, California) (SIGGRAPH '04). 630\u2013634."},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201387"},{"key":"e_1_2_2_11_1","first-page":"5","article-title":"1995","volume":"16","author":"Byrd R. H.","year":"1995","unstructured":"R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. 1995. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM J. Sci. Comp. 16, 5 (1995), 1190\u20131208.","journal-title":"A Limited Memory Algorithm for Bound Constrained Optimization. SIAM J. Sci. Comp."},{"key":"e_1_2_2_12_1","first-page":"10","article-title":"2023. Optimizing contact patterns for robot locomotion via geometric mechanics","volume":"42","author":"Chong B.","year":"2023","unstructured":"B. Chong, T. Wang, L. Bo, S. Li, P. C. Muthukrishnan, J. He, D. Irvine, H. Choset, G. Blekherman, and D. I. Goldman. 2023. Optimizing contact patterns for robot locomotion via geometric mechanics. I. J. Rob. Res. 42, 10 (2023), 859\u2013873.","journal-title":"I. J. Rob. Res."},{"key":"e_1_2_2_13_1","first-page":"4","article-title":"2012. Deformable objects alive! ACM Trans","volume":"31","author":"Coros S.","year":"2012","unstructured":"S. Coros, S. Martin, B. Thomaszewski, C. Schumacher, R. Sumner, and M. Gross. 2012. Deformable objects alive! ACM Trans. Graph. 31, 4, Article 69 (2012), 9 pages.","journal-title":"Graph."},{"key":"e_1_2_2_14_1","first-page":"6","article-title":"2023","volume":"42","author":"Dodik A.","year":"2023","unstructured":"A. Dodik, O. Stein, V. Sitzmann, and J. Solomon. 2023. Variational Barycentric Coordinates. ACM Trans. Graph. 42, 6, Article 255 (2023), 16 pages.","journal-title":"Variational Barycentric Coordinates. ACM Trans. Graph."},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2021.3070305"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/3490168"},{"key":"e_1_2_2_17_1","unstructured":"L. Euler. 1744. Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Bousquet Lausanne & Geneva. English translation at WikiSource."},{"key":"e_1_2_2_18_1","doi-asserted-by":"crossref","unstructured":"T. Frankel. 2011. The Geometry of Physics: An Introduction (3 ed.). CUP.","DOI":"10.1017\/CBO9781139061377"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/3414685.3417766"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.91.023012"},{"key":"e_1_2_2_21_1","first-page":"4","article-title":"2022","volume":"41","author":"Goel P.","year":"2022","unstructured":"P. Goel and D. L. James. 2022. Unified Many-Worlds Browsing of Arbitrary Physics-Based Animations. ACM Trans. Graph. 41, 4 (2022), 156:1\u201315.","journal-title":"Unified Many-Worlds Browsing of Arbitrary Physics-Based Animations. ACM Trans. Graph."},{"key":"e_1_2_2_22_1","first-page":"4","article-title":"1955","volume":"32","author":"Gray J.","year":"1955","unstructured":"J. Gray and G. J. Hancock. 1955. The Propulsion of Sea-Urchin Spermatozoa. J. Exp. Biol. 32, 4 (1955), 802\u2013814.","journal-title":"The Propulsion of Sea-Urchin Spermatozoa. J. Exp. Biol."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592417"},{"key":"e_1_2_2_24_1","volume-title":"Proceedings of the IEEE\/CVF International Conference on Computer Vision. 14181\u201314191","author":"Hartman E.","unstructured":"E. Hartman, E. Pierson, M. Bauer, N. Charon, and M. Daoudi. 2023a. Bare-esa: A riemannian framework for unregistered human body shapes. In Proceedings of the IEEE\/CVF International Conference on Computer Vision. 14181\u201314191."},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-022-01743-0"},{"key":"e_1_2_2_26_1","unstructured":"F. Hartwig O. Gross M. Rumpf and P. Schr\u00f6der. 2025. Optimal motion from shape change. Oberwol fach Reports 5\/2025 15\u201318."},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392480"},{"key":"e_1_2_2_28_1","volume-title":"Proceedings of the Symposium on Geometry Processing","author":"Heeren B.","unstructured":"B. Heeren, M. Rumpf, P. Schr\u00f6der, M. Wardetzky, and B. Wirth. 2014. Exploring the Geometry of the Space of Shells. In Proceedings of the Symposium on Geometry Processing (Cardiff, UK) (SGP '14). Eurographics Association, 247\u2013256."},{"key":"e_1_2_2_29_1","doi-asserted-by":"crossref","unstructured":"B. Heeren M. Rumpf P. Schr\u00f6der M. Wardetzky and B. Wirth. 2016. Splines in the Space of Shells. Comp. Graph. Forum (2016).","DOI":"10.1111\/cgf.12968"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2012.01.001"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/2019627.2019638"},{"key":"e_1_2_2_32_1","volume-title":"Biologically Inspired Robots (Snake-like Locomotor and Manipulator)","author":"Hirose S.","unstructured":"S. Hirose. 1993. Biologically Inspired Robots (Snake-like Locomotor and Manipulator). Oxford University Press."},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1145\/218380.218414"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1145\/3657648"},{"key":"e_1_2_2_35_1","first-page":"3","article-title":"2023","volume":"6","author":"Jia S.","year":"2023","unstructured":"S. Jia, S. Wang, T.-M. Li, and A. Chern. 2023. Physical Cyclic Animations. Proc. ACM Comput. Graph. Interact. Tech. 6, 3, Article 45 (2023), 18 pages.","journal-title":"Physical Cyclic Animations. Proc. ACM Comput. Graph. Interact. Tech."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276466"},{"key":"e_1_2_2_37_1","first-page":"18","article-title":"2016","volume":"49","author":"Kadam S.","year":"2016","unstructured":"S. Kadam and R. N. Banavar. 2016. Geometric Controllability of The Purcell's Swimmer and its Symmetrized Cousin. IFAC-PapersOnLine 49, 18 (2016), 988\u2013993. 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016.","journal-title":"IFAC-PapersOnLine"},{"key":"e_1_2_2_38_1","unstructured":"S. Kang X. Xu J. Sarva L. Liang and H. Yang. 2024. Fast and Certifiable Trajectory Optimization. arXiv:2406.05846"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/2366145.2366215"},{"key":"e_1_2_2_40_1","volume-title":"torchcubicspline. https:\/\/github.com\/patrick-kidger\/torchcubicspline Last accessed","author":"Kidger P.","year":"2025","unstructured":"P. Kidger. 2017. torchcubicspline. https:\/\/github.com\/patrick-kidger\/torchcubicspline Last accessed 17 January 2025."},{"key":"e_1_2_2_41_1","first-page":"2","article-title":"2009","volume":"28","author":"Kobilarov M.","year":"2009","unstructured":"M. Kobilarov, K. Crane, and M. Desbrun. 2009. Lie Group Integrators for Animation and Control of Vehicles. ACM Trans. Graph. 28, 2 (2009), 16:1\u201314.","journal-title":"ACM Trans. Graph."},{"key":"e_1_2_2_42_1","first-page":"1","article-title":"2010","volume":"16","author":"Kwatra N.","year":"2010","unstructured":"N. Kwatra, C. Wojtan, M. Carlson, I. A. Essa, P. J. Mucha, and G. Turk. 2010. Fluid Simulation with Articulated Bodies. IEEE Trans. Vis. Comp. Graph. 16, 1 (2010), 70\u201380.","journal-title":"Fluid Simulation with Articulated Bodies. IEEE Trans. Vis. Comp. Graph."},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-70529-1_154"},{"key":"e_1_2_2_44_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392425"},{"key":"e_1_2_2_45_1","first-page":"31","article-title":"2022. Robotic swimming in curved space via geometric phase","volume":"119","author":"Li S.","year":"2022","unstructured":"S. Li, T. Wang, V. H. Kojouharov, J. McInerney, E. Aydin, Y. Ozkan-Aydin, D. I. Goldman, and D. Z. Rocklin. 2022. Robotic swimming in curved space via geometric phase. Proc. Nat. Acad. Sci. 119, 31 (2022), e2200924119.","journal-title":"Proc. Nat. Acad. Sci."},{"key":"e_1_2_2_46_1","volume-title":"SMPL: A Skinned Multi-Person Linear Model (1 ed.)","author":"Loper M.","year":"2023","unstructured":"M. Loper, N. Mahmood, J. Romero, G. Pons-Moll, and M. J. Black. 2023. SMPL: A Skinned Multi-Person Linear Model (1 ed.). Association for Computing Machinery, New York, NY, USA."},{"key":"e_1_2_2_47_1","doi-asserted-by":"crossref","unstructured":"J. E. Marsden and T. S. Ratiu. 1999. Introduction to Mechanics and Symmetry (2 ed.). Springer New York NY.","DOI":"10.1007\/978-0-387-21792-5"},{"key":"e_1_2_2_48_1","first-page":"357","article-title":"2001","volume":"10","author":"Marsden J. E.","year":"2001","unstructured":"J. E. Marsden and M. West. 2001. Discrete Mechanics and Variational Integrators. Act. Num. 10 (May 2001), 357\u2013514.","journal-title":"Discrete Mechanics and Variational Integrators. Act. Num."},{"key":"e_1_2_2_49_1","volume-title":"ACM SIGGRAPH 2011 Papers. Association for Computing Machinery","author":"Martin S.","unstructured":"S. Martin, B. Thomaszewski, E. Grinspun, and M. Gross. 2011. Example-based elastic materials. In ACM SIGGRAPH 2011 Papers. Association for Computing Machinery, New York, NY, USA, Article 72, 8 pages."},{"key":"e_1_2_2_50_1","first-page":"6","article-title":"2019","volume":"38","author":"Min S.","year":"2019","unstructured":"S. Min, J. Won, S. Lee, J. Park, and J. Lee. 2019. SoftCon: Simulation and Control of Soft-Bodied Animals with Biomimetic Actuators. ACM Trans. Graph. 38, 6 (2019), 208:1\u201312.","journal-title":"ACM Trans. Graph."},{"key":"e_1_2_2_51_1","unstructured":"R. Montgomery. 2002. A tour of subriemannian geometries their geodesics and applications. Number 91 in Mathematical surveys and monographs. Amer. Math. Soc."},{"key":"e_1_2_2_52_1","first-page":"6","article-title":"2024","volume":"43","author":"Nabizadeh M. S.","year":"2024","unstructured":"M. S. Nabizadeh, R. Roy-Chowdhury, H. Yin, R. Ramamoorthi, and A. Chern. 2024. Fluid Implicit Particles on Coadjoint Orbits. ACM Trans. Graph. 43, 6, Article 270 (2024), 38 pages.","journal-title":"Fluid Implicit Particles on Coadjoint Orbits. ACM Trans. Graph."},{"key":"e_1_2_2_53_1","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2024.3425448"},{"key":"e_1_2_2_54_1","first-page":"7","article-title":"1998","volume":"17","author":"Ostrowski J.","year":"1998","unstructured":"J. Ostrowski and J. Burdick. 1998. The Geometric Mechanics of Undulatory Robotic Locomotion. I. J. Rob. Res. 17, 7 (1998), 683\u2013701.","journal-title":"The Geometric Mechanics of Undulatory Robotic Locomotion. I. J. Rob. Res."},{"key":"e_1_2_2_55_1","first-page":"4","article-title":"2019","volume":"38","author":"Padilla M.","year":"2019","unstructured":"M. Padilla, A. Chern, F. Kn\u00f6ppel, U. Pinkall, and P. Schr\u00f6der. 2019. On Bubble Rings and Ink Chandeliers. ACM Trans. Graph. 38, 4 (2019), 129:1\u201314.","journal-title":"On Bubble Rings and Ink Chandeliers. ACM Trans. Graph."},{"key":"e_1_2_2_56_1","doi-asserted-by":"publisher","DOI":"10.1145\/3658198"},{"key":"e_1_2_2_57_1","first-page":"3","article-title":"2018","volume":"37","author":"Pan Z.","year":"2018","unstructured":"Z. Pan and D. Manocha. 2018. Active Animations of Reduced Deformable Models with Environment Interactions. ACM Trans. Graph. 37, 3, Article 36 (2018), 17 pages.","journal-title":"ACM Trans. Graph."},{"key":"e_1_2_2_58_1","volume-title":"J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. K\u00f6pf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala.","author":"Paszke A.","year":"2019","unstructured":"A. Paszke, S. Gross, F. Massa, A: Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. K\u00f6pf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala. 2019. PyTorch: an imperative style, high-performance deep learning library. Curran Associates Inc., Red Hook, NY, USA."},{"key":"e_1_2_2_59_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-39838-4"},{"key":"e_1_2_2_60_1","first-page":"1","article-title":"1993","volume":"2","author":"Pinkall U.","year":"1993","unstructured":"U. Pinkall and K. Polthier. 1993. Computing Discrete Minimal Surfaces and Their Conjugates. Exp. Math. 2, 1 (1993), 15\u201336.","journal-title":"Computing Discrete Minimal Surfaces and Their Conjugates. Exp. Math."},{"key":"e_1_2_2_61_1","volume-title":"Life at Low Reynolds Number. Amer. J. Phys. 45, 3","author":"Purcell E. M.","year":"1977","unstructured":"E. M. Purcell. 1977. Life at Low Reynolds Number. Amer. J. Phys. 45, 3 (1977), 3\u201311."},{"key":"e_1_2_2_62_1","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555454"},{"key":"e_1_2_2_63_1","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2019.2915424"},{"key":"e_1_2_2_64_1","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.2009.5152817"},{"key":"e_1_2_2_65_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2320517121"},{"key":"e_1_2_2_66_1","first-page":"4","article-title":"2024","volume":"43","author":"Sassen J.","year":"2024","unstructured":"J. Sassen, H. Schumacher, M. Rumpf, and K. Crane. 2024. Repulsive Shells. ACM Trans. Graph. 43, 4 (2024).","journal-title":"Repulsive Shells. ACM Trans. Graph."},{"key":"e_1_2_2_67_1","first-page":"10","article-title":"2007. Geometric motion planning analysis for two classes of underactuated mechanical systems","volume":"26","author":"Shammas E. A.","year":"2007","unstructured":"E. A. Shammas, H. Choset, and A. A. Rizzi. 2007. Geometric motion planning analysis for two classes of underactuated mechanical systems. I. J. Rob. Res. 26, 10 (2007), 1043\u20131073.","journal-title":"I. J. Rob. Res."},{"key":"e_1_2_2_68_1","doi-asserted-by":"publisher","DOI":"10.1017\/S002211208900025X"},{"key":"e_1_2_2_69_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112089000261"},{"key":"e_1_2_2_70_1","first-page":"6","article-title":"1989c","volume":"57","author":"Shapere A.","year":"1989","unstructured":"A. Shapere and F. Wilczek. 1989c. Gauge Kinematics of Deformable Bodies. Amer. J. Phys. 57, 6 (1989), 514\u2013518.","journal-title":"Gauge Kinematics of Deformable Bodies. Amer. J. Phys."},{"key":"e_1_2_2_71_1","first-page":"4","article-title":"2024a","volume":"43","author":"Soliman Y.","year":"2024","unstructured":"Y. Soliman, M. Padilla, O. Gross, F. Kn\u00f6ppel, U. Pinkall, and P. Schr\u00f6der. 2024a. Going with the Flow. ACM Trans. Graph. 43, 4 (2024), 57:1\u201312.","journal-title":"Going with the Flow. ACM Trans. Graph."},{"key":"e_1_2_2_72_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.difgeo.2024.102200"},{"key":"e_1_2_2_73_1","unstructured":"O. Sorkine and M. Alexa. 2007. As-Rigid-As-Possible Surface Modeling. In Geometry Processing Alexander Belyaev and Michael Garland (Eds.). The Eurographics Association."},{"key":"e_1_2_2_74_1","first-page":"068105","article-title":"2007","volume":"98","author":"Tam D.","year":"2007","unstructured":"D. Tam and A. E. Hosoi. 2007. Optimal Stroke Patterns for Purcell's Three-Link Swimmer. Phys. R. Lett. 98 (Feb 2007), 068105. Issue 6.","journal-title":"Optimal Stroke Patterns for Purcell's Three-Link Swimmer. Phys. R. Lett."},{"key":"e_1_2_2_75_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964953"},{"key":"e_1_2_2_76_1","volume-title":"Wissenschaftliche Abhandlungen.","author":"von Helmholtz H.","unstructured":"H. von Helmholtz. 1882. Zur Theorie der station\u00e4ren Str\u00f6me in reibenden Fl\u00fcssigkeiten. In Wissenschaftliche Abhandlungen. Vol. I. J. A. Barth, 223\u2013230."},{"key":"e_1_2_2_77_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531366"},{"key":"e_1_2_2_78_1","first-page":"4","article-title":"2012","volume":"31","author":"Wei\u00dfmann S.","year":"2012","unstructured":"S. Wei\u00dfmann and U. Pinkall. 2012. Underwater Rigid Body Dynamics. ACM Trans. Graph. 31, 4 (2012), 104:1\u20137.","journal-title":"Underwater Rigid Body Dynamics. ACM Trans. Graph."},{"key":"e_1_2_2_79_1","first-page":"4","article-title":"1988. Spacetime constraints","volume":"22","author":"Witkin A.","year":"1988","unstructured":"A. Witkin and M. Kass. 1988. Spacetime constraints. SIGGRAPH Comput. Graph. 22, 4 (1988), 159\u2013168.","journal-title":"SIGGRAPH Comput. Graph."},{"key":"e_1_2_2_80_1","first-page":"1","article-title":"1996","volume":"15","author":"Wooten W. L.","year":"1996","unstructured":"W. L. Wooten and J. K. Hodgins. 1996. Animation of Human Diving. Comp. Graph. Forum 15, 1 (1996), 3\u201313.","journal-title":"Animation of Human Diving. Comp. Graph. Forum"},{"key":"e_1_2_2_81_1","first-page":"3","article-title":"2003","volume":"22","author":"Wu J.-C.","year":"2003","unstructured":"J.-C. Wu and Z. Popovi\u0107. 2003. Realistic Modeling of Bird Flight Animations. ACM Trans. Graph. 22, 3 (2003), 888\u2013895.","journal-title":"Realistic Modeling of Bird Flight Animations. ACM Trans. Graph."},{"key":"e_1_2_2_82_1","volume-title":"2022 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS). 12598\u201312605","author":"Zhang J. Z.","unstructured":"J. Z. Zhang, Y. Zhang, P. Ma, E. Nava, T. Du, P. Arm, W. Matusik, and R. K. Katzschmann. 2022. Sim2Real for Soft Robotic Fish via Differentiable Simulation. In 2022 IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS). 12598\u201312605."},{"key":"e_1_2_2_83_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4898629"},{"key":"e_1_2_2_84_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2113222119"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3731187","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T17:51:36Z","timestamp":1774633896000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3731187"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,27]]},"references-count":84,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8,1]]}},"alternative-id":["10.1145\/3731187"],"URL":"https:\/\/doi.org\/10.1145\/3731187","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,27]]},"assertion":[{"value":"2025-07-27","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}