{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T02:04:14Z","timestamp":1780452254794,"version":"3.54.1"},"reference-count":82,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T00:00:00Z","timestamp":1776211200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T00:00:00Z","timestamp":1780444800000},"content-version":"vor","delay-in-days":49,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-1436636"],"award-info":[{"award-number":["CMMI-1436636"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1427193"],"award-info":[{"award-number":["IIS-1427193"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Mitsui USA Foundation"},{"DOI":"10.13039\/100006732","name":"New York University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100006732","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100014188","name":"Ministry of Science and ICT, South Korea","doi-asserted-by":"publisher","award":["RS-2024-00421519"],"award-info":[{"award-number":["RS-2024-00421519"]}],"id":[{"id":"10.13039\/501100014188","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Intell Robot Syst"],"DOI":"10.1007\/s10846-026-02372-2","type":"journal-article","created":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T17:13:29Z","timestamp":1776273209000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Robot Trajectory Direct Collocation with Conditional Constraints for Optimal Phase Transitions"],"prefix":"10.1007","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3612-3315","authenticated-orcid":false,"given":"William Z.","family":"Peng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3470-8221","authenticated-orcid":false,"given":"Hyunjong","family":"Song","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0305-5405","authenticated-orcid":false,"given":"Joo H.","family":"Kim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,15]]},"reference":[{"key":"2372_CR1","unstructured":"Wieber, P. B.: On the stability of walking systems. In: Proceedings of the international workshop on humanoid and human friendly robotics, Tsukuba, Japan (2002)"},{"key":"2372_CR2","unstructured":"Gabiccini, M., Artoni, A., Pannocchia, G., Gillis, J.: A computational framework for environment-aware robotic manipulation planning. International Symposium on Robotics Research (ISRR), Sestri Levante, Italy (2015)"},{"issue":"16","key":"2372_CR3","doi-asserted-by":"publisher","first-page":"1825","DOI":"10.1080\/01691864.2012.685259","volume":"26","author":"CN Cho","year":"2012","unstructured":"Cho, C.N., Kim, J.H., Kim, Y.L., Song, J.B., Kyung, J.H.: Collision detection algorithm to distinguish between intended contact and unexpected collision. Adv. Robot. 26(16), 1825\u20131840 (2012). https:\/\/doi.org\/10.1080\/01691864.2012.685259","journal-title":"Adv. Robot."},{"key":"2372_CR4","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.mechmachtheory.2013.02.001","volume":"64","author":"JH Kim","year":"2013","unstructured":"Kim, J.H., Joo, C.B.: Optimal motion planning of redundant manipulators with controlled task infeasibility. Mech. Mach. Theory 64, 155\u2013174 (2013). https:\/\/doi.org\/10.1016\/j.mechmachtheory.2013.02.001","journal-title":"Mech. Mach. Theory"},{"key":"2372_CR5","doi-asserted-by":"publisher","unstructured":"Joo, C. B., Kim, J. H.: SQP-based mobile manipulator motion planning with controlled infeasibility for physically valid task failure. ASME IDETC\/CIE 37th Mechanisms and Robotics Conference, Portland, OR, USA. (2013). https:\/\/doi.org\/10.1115\/DETC2013-13377","DOI":"10.1115\/DETC2013-13377"},{"issue":"12","key":"2372_CR6","doi-asserted-by":"publisher","first-page":"1312","DOI":"10.1177\/0278364917733549","volume":"36","author":"S Mitsch","year":"2017","unstructured":"Mitsch, S., Ghorbal, K., Vogelbacher, D., Platzer, A.: Formal verification of obstacle avoidance and navigation of ground robots. Int. J. Robot. Res. 36(12), 1312\u20131340 (2017). https:\/\/doi.org\/10.1177\/0278364917733549","journal-title":"Int. J. Robot. Res."},{"issue":"2","key":"2372_CR7","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1109\/LCSYS.2020.3002852","volume":"5","author":"B Sangiovanni","year":"2021","unstructured":"Sangiovanni, B., Incremona, G.P., Piastra, M., Ferrara, A.: Self-configuring robot path planning with obstacle avoidance via deep reinforcement learning. IEEE Control Syst. Lett. 5(2), 397\u2013402 (2021). https:\/\/doi.org\/10.1109\/LCSYS.2020.3002852","journal-title":"IEEE Control Syst. Lett."},{"key":"2372_CR8","doi-asserted-by":"publisher","unstructured":"Gajamohan, M., Muehlebach, M., Widmer, T., D\u2019Andrea, R.: The Cubli: A reaction wheel based 3D inverted pendulum. IEEE European Control Conference (ECC), Zurich, Switzerland. (2013). https:\/\/doi.org\/10.23919\/ecc.2013.6669562.","DOI":"10.23919\/ecc.2013.6669562"},{"issue":"4","key":"2372_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10846-024-02193-1","volume":"110","author":"F Tassi","year":"2024","unstructured":"Tassi, F., Ajoudani, A.: Multi-modal and adaptive robot control through hierarchical quadratic programming. J. Intell. Robot. Syst. 110(4), 1\u201317 (2024). https:\/\/doi.org\/10.1007\/s10846-024-02193-1","journal-title":"J. Intell. Robot. Syst."},{"key":"2372_CR10","doi-asserted-by":"crossref","unstructured":"Reher, J. P., Hereid, A., Kolathaya, S., Hubicki, C. M., Ames, A. D.: Algorithmic foundations of realizing multi-contact locomotion on the humanoid robot DURUS. In: Goldberg, K., Abbeel, P., Bekris, K., and Miller, L. (eds.) (Chapter 26) in Algorithmic Foundations of Robotics XII: Proceedings of the Twelfth Workshop on the Algorithmic Foundations of Robotics, pp. 400\u2013415. Springer Nature, Cham, Switzerland (2020)","DOI":"10.1007\/978-3-030-43089-4_26"},{"key":"2372_CR11","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1007\/BF02071065","volume":"37","author":"O Stryk","year":"1992","unstructured":"Stryk, O., Bulirsch, R.: Direct and indirect methods for trajectory optimization. Ann. Oper. Res. 37, 357\u2013373 (1992)","journal-title":"Ann. Oper. Res."},{"key":"2372_CR12","volume-title":"Practical methods for optimal control using nonlinear programming","author":"JT Betts","year":"2001","unstructured":"Betts, J.T.: Practical methods for optimal control using nonlinear programming, 2nd ed. SIAM, Philadelphia, PA, USA (2001)","edition":"2nd ed."},{"key":"2372_CR13","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1109\/TRO.2023.3324580","volume":"40","author":"PM Wensing","year":"2024","unstructured":"Wensing, P.M., Posa, M., Hu, Y., Escande, A., Mansard, N., Del Prete, A.: Optimization-based control for dynamic legged robots. IEEE Trans. Robot. 40, 43\u201363 (2024). https:\/\/doi.org\/10.1109\/TRO.2023.3324580","journal-title":"IEEE Trans. Robot."},{"issue":"2","key":"2372_CR14","doi-asserted-by":"publisher","first-page":"193","DOI":"10.2514\/2.4231","volume":"21","author":"JT Betts","year":"1998","unstructured":"Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control. Dyn. 21(2), 193\u2013207 (1998). https:\/\/doi.org\/10.2514\/2.4231","journal-title":"J. Guid. Control. Dyn."},{"key":"2372_CR15","doi-asserted-by":"crossref","unstructured":"Pont\u00f3n, B., Schaal, S., Righetti, L.: On the effects of measurement uncertainty in optimal control of contact interactions. In: Goldberg, K., Abbeel, P., Bekris, K., and Miller, L. (eds.) (Chapter 50) in Algorithmic Foundations of Robotics XII: Proceedings of the Twelfth Workshop on the Algorithmic Foundations of Robotics, pp. 784\u2013799. Springer Nature, Cham, Switzerland (2020)","DOI":"10.1007\/978-3-030-43089-4_50"},{"key":"2372_CR16","doi-asserted-by":"publisher","unstructured":"Kurtz, V., Lin, H.: Contact-implicit trajectory optimization with hydroelastic contact and iLQR. IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Kyoto, Japan (2022). https:\/\/doi.org\/10.1109\/IROS47612.2022.9981686","DOI":"10.1109\/IROS47612.2022.9981686"},{"issue":"4","key":"2372_CR17","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1137\/16M1062569","volume":"59","author":"M Kelly","year":"2017","unstructured":"Kelly, M.: An introduction to trajectory optimization: how to do your own direct collocation. SIAM Rev. 59(4), 849\u2013904 (2017). https:\/\/doi.org\/10.1137\/16M1062569","journal-title":"SIAM Rev."},{"key":"2372_CR18","volume-title":"Differential dynamic programming","author":"DH Jacobson","year":"1970","unstructured":"Jacobson, D.H., Mayne, D.Q.: Differential dynamic programming. American Elsevier Publishing Company, New York, NY, USA (1970)"},{"key":"2372_CR19","unstructured":"Bertsekas, D. P.: Dynamic programming and optimal control, 3rd ed., Vol. 1, Athena Scientific, Belmont, Massachusetts, USA (1996)"},{"key":"2372_CR20","volume-title":"Optimal control theory: an introduction","author":"DE Kirk","year":"1998","unstructured":"Kirk, D.E.: Optimal control theory: an introduction. Prentice-Hall, Englewood Cliffs, NJ, USA (1998)"},{"key":"2372_CR21","doi-asserted-by":"publisher","unstructured":"Kajita, S., Sakaguchi, T., Nakaoka, S., Morisawa, M., Kaneko, K., and Kanehiro, F.: Quick squatting motion generation of a humanoid robot for falling damage reduction. IEEE International Conference on Cyborg and Bionic Systems (CBS), Beijing, China (2017). https:\/\/doi.org\/10.1109\/CBS.2017.8266127","DOI":"10.1109\/CBS.2017.8266127"},{"key":"2372_CR22","doi-asserted-by":"publisher","unstructured":"Pardo, D., Neunert, M., Winkler, A., Grandia, R., Buchli, J.: Hybrid direct collocation and control in the constraint-consistent subspace for dynamic legged robot locomotion. Robotics: Science and Systems, Cambridge, MA, USA (2017). https:\/\/doi.org\/10.15607\/rss.2017.xiii.042","DOI":"10.15607\/rss.2017.xiii.042"},{"issue":"2","key":"2372_CR23","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1109\/TRO.2017.2783371","volume":"34","author":"A Hereid","year":"2018","unstructured":"Hereid, A., Hubicki, C.M., Cousineau, E.A., Ames, A.D.: Dynamic humanoid locomotion: a scalable formulation for HZD gait optimization. IEEE Trans. Robot. 34(2), 370\u2013387 (2018). https:\/\/doi.org\/10.1109\/TRO.2017.2783371","journal-title":"IEEE Trans. Robot."},{"issue":"1","key":"2372_CR24","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1109\/TRO.2022.3193776","volume":"39","author":"R Bordalba","year":"2023","unstructured":"Bordalba, R., Schoels, T., Ros, L., Porta, J.M., Diehl, M.: Direct collocation methods for trajectory optimization in constrained robotic systems. IEEE Trans. Robot. 39(1), 183\u2013202 (2023). https:\/\/doi.org\/10.1109\/TRO.2022.3193776","journal-title":"IEEE Trans. Robot."},{"issue":"1","key":"2372_CR25","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1002\/nme.3162","volume":"88","author":"JH Kim","year":"2011","unstructured":"Kim, J.H., Abdel-Malek, K., Xiang, Y., Yang, J.J., Arora, J.S.: Concurrent motion planning and reaction load distribution for redundant dynamic systems under external holonomic constraints. Int. J. Numer. Methods Eng. 88(1), 47\u201365 (2011). https:\/\/doi.org\/10.1002\/nme.3162","journal-title":"Int. J. Numer. Methods Eng."},{"key":"2372_CR26","doi-asserted-by":"publisher","unstructured":"Mummolo, C., Mangialardi, L., Kim, J. H.: Contact status optimization of multibody dynamic systems using dual variable transformation. Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC\/CIE, Buffalo, New York, USA, (2014). https:\/\/doi.org\/10.1115\/DETC2014-34193.","DOI":"10.1115\/DETC2014-34193"},{"key":"2372_CR27","doi-asserted-by":"publisher","unstructured":"Mummolo, C., Mangialardi, L., Kim, J. H.: Concurrent contact planning and trajectory optimization in one step walking motion. Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC\/CIE, Boston, Massachusetts, USA (2015). https:\/\/doi.org\/10.1115\/DETC2015-47745","DOI":"10.1115\/DETC2015-47745"},{"key":"2372_CR28","doi-asserted-by":"publisher","unstructured":"Posa, M., Kuindersma, S., Tedrake, R.: Optimization and stabilization of trajectories for constrained dynamical systems. IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden (2016). https:\/\/doi.org\/10.1109\/ICRA.2016.7487270","DOI":"10.1109\/ICRA.2016.7487270"},{"issue":"4","key":"2372_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10846-023-02013-y","volume":"109","author":"A Arefeen","year":"2023","unstructured":"Arefeen, A., Quarnstrom, J., Syed, S.P.Q., Bai, H., Xiang, Y.: Human\u2013robot collaborative lifting motion prediction and experimental validation. J. Intell. Robot. Syst. 109(4), 1\u201316 (2023). https:\/\/doi.org\/10.1007\/s10846-023-02013-y","journal-title":"J. Intell. Robot. Syst."},{"key":"2372_CR30","doi-asserted-by":"crossref","unstructured":"Pratt, J. E., Tedrake, R.: Velocity-based stability margins for fast bipedal walking. In: Diehl, M., and Mombaur, K. (eds.) (Chapter 14) in Fast Motions in Biomechanics and Robotics: Optimization and Feedback Control, pp. 299\u2013324. Springer-Verlag, Berlin Heidelberg, Germany (2006)","DOI":"10.1007\/978-3-540-36119-0_14"},{"issue":"2","key":"2372_CR31","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1109\/LCSYS.2019.2960442","volume":"4","author":"BS Banjanin","year":"2020","unstructured":"Banjanin, B.S., Burden, S.A.: Nonsmooth optimal value and policy functions in mechanical systems subject to unilateral constraints. IEEE Control Syst. Lett. 4(2), 506\u2013511 (2020). https:\/\/doi.org\/10.1109\/LCSYS.2019.2960442","journal-title":"IEEE Control Syst. Lett."},{"issue":"10","key":"2372_CR32","doi-asserted-by":"publisher","first-page":"1445","DOI":"10.1002\/nme.1666","volume":"67","author":"LP Ellekilde","year":"2006","unstructured":"Ellekilde, L.P., Petersen, H.G.: Simulating non-holonomic constraints within the LCP-based simulation framework. Int. J. Numer. Methods Eng. 67(10), 1445\u20131466 (2006). https:\/\/doi.org\/10.1002\/nme.1666","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"4","key":"2372_CR33","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1016\/j.mechmachtheory.2010.11.019","volume":"46","author":"JH Kim","year":"2011","unstructured":"Kim, J.H.: Optimization of throwing motion planning for whole-body humanoid mechanism: sidearm and maximum distance. Mech. Mach. Theory 46(4), 438\u2013453 (2011). https:\/\/doi.org\/10.1016\/j.mechmachtheory.2010.11.019","journal-title":"Mech. Mach. Theory"},{"issue":"3","key":"2372_CR34","doi-asserted-by":"publisher","first-page":"60:1","DOI":"10.1145\/1531326.1531366","volume":"28","author":"K Wampler","year":"2009","unstructured":"Wampler, K., Popovi\u0107, Z.: Optimal gait and form for animal locomotion. ACM Trans. Graph. 28(3), 60:1-60:8 (2009). https:\/\/doi.org\/10.1145\/1531326.1531366","journal-title":"ACM Trans. Graph."},{"key":"2372_CR35","doi-asserted-by":"publisher","unstructured":"Shield, S., Patel, A.: Balancing stability and maneuverability during rapid gait termination in fast biped robots,\u201d IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada (2017). https:\/\/doi.org\/10.1109\/IROS.2017.8206319","DOI":"10.1109\/IROS.2017.8206319"},{"issue":"9","key":"2372_CR36","doi-asserted-by":"publisher","first-page":"1140","DOI":"10.1177\/0278364915612572","volume":"35","author":"W Xi","year":"2016","unstructured":"Xi, W., Yesilevskiy, Y., Remy, C.D.: Selecting gaits for economical locomotion of legged robots. Int. J. Robot. Res. 35(9), 1140\u20131154 (2016). https:\/\/doi.org\/10.1177\/0278364915612572","journal-title":"Int. J. Robot. Res."},{"issue":"3","key":"2372_CR37","doi-asserted-by":"publisher","first-page":"036002:1","DOI":"10.1088\/1748-3190\/aaaa9e","volume":"13","author":"Y Yesilevskiy","year":"2018","unstructured":"Yesilevskiy, Y., Yang, W., Remy, C.D.: Spine morphology and energetics: how principles from nature apply to robotics. Bioinspir. Biomim. 13(3), 036002:1-036002:13 (2018). https:\/\/doi.org\/10.1088\/1748-3190\/aaaa9e","journal-title":"Bioinspir. Biomim."},{"issue":"3","key":"2372_CR38","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1023\/A:1008292328909","volume":"14","author":"M Anitescu","year":"1997","unstructured":"Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14(3), 231\u2013247 (1997). https:\/\/doi.org\/10.1023\/A:1008292328909","journal-title":"Nonlinear Dyn."},{"key":"2372_CR39","doi-asserted-by":"publisher","unstructured":"Tassa, Y., Todorov, E.: Stochastic complementarity for local control of discontinuous dynamics. Robotics: Science and Systems, Zaragoza, Spain (2010). https:\/\/doi.org\/10.15607\/RSS.2010.VI.022","DOI":"10.15607\/RSS.2010.VI.022"},{"issue":"3","key":"2372_CR40","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1109\/LRA.2018.2798285","volume":"3","author":"AW Winkler","year":"2018","unstructured":"Winkler, A.W., Bellicoso, C.D., Hutter, M., Buchli, J.: Gait and trajectory optimization for legged systems through phase-based end-effector parameterization. IEEE Robot. Autom. Lett. 3(3), 1560\u20131567 (2018). https:\/\/doi.org\/10.1109\/LRA.2018.2798285","journal-title":"IEEE Robot. Autom. Lett."},{"issue":"3","key":"2372_CR41","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/s10514-015-9479-3","volume":"40","author":"S Kuindersma","year":"2016","unstructured":"Kuindersma, S., Deits, R., Fallon, M., Valenzuela, A., Dai, H., Permenter, F., Koolen, T., et al.: Optimization-based locomotion planning, estimation, and control design for the atlas humanoid robot. Autonomous Robot. 40(3), 429\u2013455 (2016). https:\/\/doi.org\/10.1007\/s10514-015-9479-3","journal-title":"Autonomous Robot."},{"issue":"3","key":"2372_CR42","doi-asserted-by":"publisher","first-page":"2531","DOI":"10.1109\/LRA.2017.2779821","volume":"3","author":"B Aceituno-Cabezas","year":"2018","unstructured":"Aceituno-Cabezas, B., Mastalli, C., Dai, H., Focchi, M., Radulescu, A., Caldwell, D.G., Cappelletto, J., et al.: Simultaneous contact, gait, and motion planning for robust multilegged locomotion via mixed-integer convex optimization. IEEE Robot. Automation Lett. 3(3), 2531\u20132538 (2018). https:\/\/doi.org\/10.1109\/LRA.2017.2779821","journal-title":"IEEE Robot. Automation Lett."},{"key":"2372_CR43","doi-asserted-by":"publisher","unstructured":"Zhang, C., Shah, J. A.: Co-optimizing task and motion planning. IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea (2016). https:\/\/doi.org\/10.1109\/IROS.2016.7759698","DOI":"10.1109\/IROS.2016.7759698"},{"issue":"2","key":"2372_CR44","doi-asserted-by":"publisher","first-page":"2966","DOI":"10.1109\/LRA.2020.2974684","volume":"5","author":"S Zimmermann","year":"2020","unstructured":"Zimmermann, S., Hakimifard, G., Zamora, M., Poranne, R., Coros, S.: A multi-level optimization framework for simultaneous grasping and motion planning. IEEE Robot. Autom. Lett. 5(2), 2966\u20132972 (2020). https:\/\/doi.org\/10.1109\/LRA.2020.2974684","journal-title":"IEEE Robot. Autom. Lett."},{"key":"2372_CR45","doi-asserted-by":"publisher","first-page":"103786\u20131","DOI":"10.1016\/j.robot.2021.103786","volume":"141","author":"A Thomas","year":"2021","unstructured":"Thomas, A., Mastrogiovanni, F., Baglietto, M.: MPTP: motion-planning-aware task planning for navigation in belief space. Robot. Auton. Syst. 141, 103786-1\u2013103818 (2021). https:\/\/doi.org\/10.1016\/j.robot.2021.103786","journal-title":"Robot. Auton. Syst."},{"key":"2372_CR46","unstructured":"Toussaint, M.: Logic-geometric programming: An optimization-based approach to combined task and motion planning. Proceedings of the International Joint Conference on Artificial Intelligence, Buenos Aires, Argentina (2015)"},{"key":"2372_CR47","doi-asserted-by":"publisher","unstructured":"Toussaint, M., Allen, K. R., Smith, K. A., Tenenbaum, J. B.: Differentiable physics and stable modes for tool-use and manipulation planning. Proceedings of Robotics: Science and Systems (RSS), Pittsburgh, PA, USA (2018). https:\/\/doi.org\/10.15607\/RSS.2018.XIV.044","DOI":"10.15607\/RSS.2018.XIV.044"},{"issue":"2","key":"2372_CR48","doi-asserted-by":"publisher","first-page":"844","DOI":"10.1109\/LRA.2020.2965875","volume":"5","author":"T Migimatsu","year":"2020","unstructured":"Migimatsu, T., Bohg, J.: Object-centric task and motion planning in dynamic environments. IEEE Robot. Autom. Lett. 5(2), 844\u2013851 (2020). https:\/\/doi.org\/10.1109\/LRA.2020.2965875","journal-title":"IEEE Robot. Autom. Lett."},{"key":"2372_CR49","doi-asserted-by":"publisher","unstructured":"Braun, C. V., Ortiz-Haro, J., Toussaint, M., Oguz, O. S.: RHH-LGP: Receding horizon and heuristics-based logic-geometric programming for task and motion planning. IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Kyoto, Japan (2022). https:\/\/doi.org\/10.1109\/IROS47612.2022.9981797","DOI":"10.1109\/IROS47612.2022.9981797"},{"issue":"81","key":"2372_CR50","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1126\/scirobotics.adg5014","volume":"8","author":"JP Sleiman","year":"2023","unstructured":"Sleiman, J.P., Farshidian, F., Hutter, M.: Versatile multicontact planning and control for legged loco-manipulation. Sci. Robot. 8(81), 1\u201316 (2023). https:\/\/doi.org\/10.1126\/scirobotics.adg5014","journal-title":"Sci. Robot."},{"issue":"4","key":"2372_CR51","doi-asserted-by":"publisher","first-page":"43:1","DOI":"10.1145\/2185520.2185539","volume":"31","author":"I Mordatch","year":"2012","unstructured":"Mordatch, I., Todorov, E., Popovi\u0107, Z.: Discovery of complex behaviors through contact-invariant optimization. ACM Trans. Graph. 31(4), 43:1-43:8 (2012). https:\/\/doi.org\/10.1145\/2185520.2185539","journal-title":"ACM Trans. Graph."},{"issue":"1","key":"2372_CR52","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1177\/0278364913506757","volume":"33","author":"M Posa","year":"2014","unstructured":"Posa, M., Cantu, C., Tedrake, R.: A direct method for trajectory optimization of rigid bodies through contact. Int. J. Robot. Res. 33(1), 69\u201381 (2014). https:\/\/doi.org\/10.1177\/0278364913506757","journal-title":"Int. J. Robot. Res."},{"issue":"2","key":"2372_CR53","doi-asserted-by":"publisher","first-page":"2242","DOI":"10.1109\/LRA.2019.2900840","volume":"4","author":"A Patel","year":"2019","unstructured":"Patel, A., Shield, S.L., Kazi, S., Johnson, A.M., Biegler, L.T.: Contact-implicit trajectory optimization using orthogonal collocation. IEEE Robot. Autom. Lett. 4(2), 2242\u20132249 (2019). https:\/\/doi.org\/10.1109\/LRA.2019.2900840","journal-title":"IEEE Robot. Autom. Lett."},{"key":"2372_CR54","doi-asserted-by":"publisher","unstructured":"Turski, M. R., Norby, J., Johnson, A. M.: Staged contact optimization: combining contact-implicit and multi-phase hybrid trajectory optimization. IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Detroit, MI, USA (2023). https:\/\/doi.org\/10.48550\/arXiv.2304.04923","DOI":"10.48550\/arXiv.2304.04923"},{"issue":"1","key":"2372_CR55","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1109\/LRA.2016.2547024","volume":"2","author":"T Marcucci","year":"2017","unstructured":"Marcucci, T., Gabiccini, M., Artoni, A.: A two-stage trajectory optimization strategy for articulated bodies with unscheduled contact sequences. IEEE Robot. Autom. Lett. 2(1), 104\u2013111 (2017). https:\/\/doi.org\/10.1109\/LRA.2016.2547024","journal-title":"IEEE Robot. Autom. Lett."},{"key":"2372_CR56","doi-asserted-by":"publisher","unstructured":"Anitescu, M.: A fixed time-step approach for multibody dynamics with contact and friction. IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Las Vegas, Nevada, USA (2003). https:\/\/doi.org\/10.1109\/iros.2003.1249734","DOI":"10.1109\/iros.2003.1249734"},{"issue":"14","key":"2372_CR57","doi-asserted-by":"publisher","first-page":"2335","DOI":"10.1002\/nme.1047","volume":"60","author":"M Anitescu","year":"2004","unstructured":"Anitescu, M., Hart, G.D.: A constraint-stabilized time-stepping approach for rigid multibody dynamics with joints, contact and friction. Int. J. Numer. Methods Eng. 60(14), 2335\u20132371 (2004). https:\/\/doi.org\/10.1002\/nme.1047","journal-title":"Int. J. Numer. Methods Eng."},{"key":"2372_CR58","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75392-6","volume-title":"Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics","author":"V Acary","year":"2008","unstructured":"Acary, V., Brogliato, B.: Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics, 1st ed. Springer-Verlag, Berlin Heidelberg, Germany (2008)","edition":"1st ed."},{"key":"2372_CR59","doi-asserted-by":"publisher","unstructured":"Xi, W., Remy, C. D.: Optimal gaits and motions for legged robots. IEEE\/RSJ International Conference on Intelligent Robots and Systems (IROS), Chicago, IL, USA (2014). https:\/\/doi.org\/10.1109\/IROS.2014.6943015","DOI":"10.1109\/IROS.2014.6943015"},{"key":"2372_CR60","volume-title":"Introduction to optimum design","author":"JS Arora","year":"2016","unstructured":"Arora, J.S.: Introduction to optimum design, 4th ed. Academic Press, Cambridge, MA, USA (2016)","edition":"4th ed."},{"issue":"3","key":"2372_CR61","doi-asserted-by":"publisher","first-page":"031103\/1","DOI":"10.1115\/1.4050095","volume":"13","author":"WZ Peng","year":"2021","unstructured":"Peng, W.Z., Song, H., Kim, J.H.: Stability region-based analysis of walking and push recovery control. J. Mech. Robot. 13(3), 031103\/1-031103\/11 (2021). https:\/\/doi.org\/10.1115\/1.4050095","journal-title":"J. Mech. Robot."},{"key":"2372_CR62","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2022.104880","volume":"174","author":"WZ Peng","year":"2022","unstructured":"Peng, W.Z., Mummolo, C., Song, H., Kim, J.H.: Whole-body balance stability regions for multi-level momentum and stepping strategies. Mech. Mach. Theory 174, 104880 (2022). https:\/\/doi.org\/10.1016\/j.mechmachtheory.2022.104880","journal-title":"Mech. Mach. Theory"},{"issue":"1","key":"2372_CR63","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1109\/TRO.2016.2623338","volume":"33","author":"S Caron","year":"2017","unstructured":"Caron, S., Pham, Q.-C., Nakamura, Y.: ZMP support areas for multi-contact mobility under frictional constraints. IEEE Trans. Robot. 33(1), 67\u201380 (2017). https:\/\/doi.org\/10.1109\/TRO.2016.2623338","journal-title":"IEEE Trans. Robot."},{"issue":"1","key":"2372_CR64","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1137\/S0036144504446096","volume":"47","author":"PE Gill","year":"2005","unstructured":"Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99\u2013131 (2005). https:\/\/doi.org\/10.1137\/S0036144504446096","journal-title":"SIAM Rev."},{"issue":"2","key":"2372_CR65","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1115\/1.4038978","volume":"10","author":"C Mummolo","year":"2018","unstructured":"Mummolo, C., Peng, W.Z., Gonzalez, C., Kim, J.H.: Contact-dependent balance stability of biped robots. J. Mech. Robot. 10(2), 1\u201313 (2018). https:\/\/doi.org\/10.1115\/1.4038978","journal-title":"J. Mech. Robot."},{"key":"2372_CR66","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511626432","volume-title":"Impact mechanics","author":"WJ Stronge","year":"2000","unstructured":"Stronge, W.J.: Impact mechanics, 1st ed. Cambridge University Press, Cambridge, UK (2000)","edition":"1st ed."},{"issue":"4","key":"2372_CR67","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1007\/s00332-016-9353-2","volume":"27","author":"C Mummolo","year":"2017","unstructured":"Mummolo, C., Mangialardi, L., Kim, J.H.: Numerical estimation of balanced and falling states for constrained legged systems. J. Nonlinear Sci. 27(4), 1291\u20131323 (2017). https:\/\/doi.org\/10.1007\/s00332-016-9353-2","journal-title":"J. Nonlinear Sci."},{"key":"2372_CR68","doi-asserted-by":"publisher","unstructured":"Peng, W. Z., Mummolo, C., Kim, J. H.: Stability criteria of balanced and steppable unbalanced states for full-body systems with implications in robotic and human gait. Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Paris, France (2020). https:\/\/doi.org\/10.1109\/ICRA40945.2020.9196820","DOI":"10.1109\/ICRA40945.2020.9196820"},{"key":"2372_CR69","doi-asserted-by":"publisher","unstructured":"Peng, W. Z., Song, H., Kim, J. H.: Reduced-order model with foot tipping allowance for legged balancing. Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC\/CIE), Virtual, Online (2021). https:\/\/doi.org\/10.1115\/DETC2021-71976","DOI":"10.1115\/DETC2021-71976"},{"key":"2372_CR70","doi-asserted-by":"publisher","unstructured":"Pratt, J., Carff, J., Drakunov, S., Goswami, A.: Capture point: A step toward humanoid push recovery. IEEE-RAS International Conference on Humanoid Robots (Humanoids), Genova, Italy (2006). https:\/\/doi.org\/10.1109\/ICHR.2006.321385","DOI":"10.1109\/ICHR.2006.321385"},{"issue":"6","key":"2372_CR71","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1103\/PhysRevE.78.066609","volume":"78","author":"M Srinivasan","year":"2008","unstructured":"Srinivasan, M., Ruina, A.: Rocking and rolling: a can that appears to rock might actually roll. Phys. Rev. E 78(6), 1\u20139 (2008). https:\/\/doi.org\/10.1103\/PhysRevE.78.066609","journal-title":"Phys. Rev. E"},{"issue":"3","key":"2372_CR72","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1080\/02681119708806242","volume":"12","author":"MJ Coleman","year":"1997","unstructured":"Coleman, M.J., Chatterjee, A., Ruina, A.: Motions of a rimless spoked wheel: a simple three-dimensional system with impacts. Dyn. Stab. Syst. 12(3), 139\u2013159 (1997). https:\/\/doi.org\/10.1080\/02681119708806242","journal-title":"Dyn. Stab. Syst."},{"issue":"2","key":"2372_CR73","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1137\/140956816","volume":"14","author":"B Gamus","year":"2015","unstructured":"Gamus, B., Or, Y.: Dynamic bipedal walking under stick-slip transitions. SIAM J. Appl. Dyn. Syst. 14(2), 609\u2013642 (2015). https:\/\/doi.org\/10.1137\/140956816","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"11\u201312","key":"2372_CR74","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1177\/02783649221110260","volume":"41","author":"JK Butterfield","year":"2022","unstructured":"Butterfield, J.K., Simha, S.N., Donelan, J.M., Collins, S.H.: The split-belt rimless wheel. Int. J. Rob. Res. 41(11\u201312), 1043\u20131076 (2022). https:\/\/doi.org\/10.1177\/02783649221110260","journal-title":"Int. J. Rob. Res."},{"issue":"2","key":"2372_CR75","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1785\/bssa0530020403","volume":"53","author":"GW Housner","year":"1963","unstructured":"Housner, G.W.: The behavior of inverted pendulum structures during earthquakes. Bull. Seismol. Soc. Am. 53(2), 403\u2013417 (1963). https:\/\/doi.org\/10.1785\/bssa0530020403","journal-title":"Bull. Seismol. Soc. Am."},{"issue":"3","key":"2372_CR76","doi-asserted-by":"publisher","first-page":"1839","DOI":"10.1007\/s11071-021-06934-x","volume":"107","author":"PL V\u00e1rkonyi","year":"2022","unstructured":"V\u00e1rkonyi, P.L., Kocsis, M., Ther, T.: Rigid impacts of three-dimensional rocking structures. Nonlinear Dyn. 107(3), 1839\u20131858 (2022). https:\/\/doi.org\/10.1007\/s11071-021-06934-x","journal-title":"Nonlinear Dyn."},{"key":"2372_CR77","unstructured":"Murray, R. M., Hauser, J.: A case study in approximate linearization: The Acrobot example. Memorandum No. UCB\/ERL M91\/46, pp. 1\u201343, (1991)"},{"issue":"1","key":"2372_CR78","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1109\/37.341864","volume":"15","author":"MW Spong","year":"1995","unstructured":"Spong, M.W.: The swing up control problem for the Acrobot. IEEE Control. Syst. Mag. 15(1), 49\u201355 (1995). https:\/\/doi.org\/10.1109\/37.341864","journal-title":"IEEE Control. Syst. Mag."},{"issue":"1","key":"2372_CR79","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1142\/S0219843604000083","volume":"1","author":"M Vukobratovi\u0107","year":"2004","unstructured":"Vukobratovi\u0107, M., Borovac, B.: Zero-moment point - Thirty five years of its life. Int. J. Humanoid Robot. 1(1), 157\u2013173 (2004). https:\/\/doi.org\/10.1142\/S0219843604000083","journal-title":"Int. J. Humanoid Robot."},{"key":"2372_CR80","doi-asserted-by":"publisher","DOI":"10.1109\/ROBOT.1999.769929","author":"A Goswami","year":"1999","unstructured":"Goswami, A.: Foot rotation indicator (FRI) point: A new gait planning tool to evaluate postural stability of biped robots. Proc. IEEE Int. Conf. Robot. Automation (ICRA) (1999). https:\/\/doi.org\/10.1109\/ROBOT.1999.769929","journal-title":"Proc. IEEE Int. Conf. Robot. Automation (ICRA)"},{"key":"2372_CR81","volume-title":"Springer tracts in advanced robotics 101: introduction to humanoid robotics","author":"S Kajita","year":"2014","unstructured":"Kajita, S., Hirukawa, H., Harada, K., Yokoi, K.: Springer tracts in advanced robotics 101: introduction to humanoid robotics. Springer-Verlag, Berlin Heidelberg, Germany (2014)"},{"issue":"4","key":"2372_CR82","doi-asserted-by":"publisher","first-page":"041003\/1","DOI":"10.1115\/1.4023698","volume":"135","author":"F Firmani","year":"2013","unstructured":"Firmani, F., Park, E.J.: Theoretical analysis of the state of balance in bipedal walking. J. Biomech. Eng. 135(4), 041003\/1-041003\/13 (2013). https:\/\/doi.org\/10.1115\/1.4023698","journal-title":"J. Biomech. Eng."}],"container-title":["Journal of Intelligent &amp; Robotic Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10846-026-02372-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10846-026-02372-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10846-026-02372-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T01:21:14Z","timestamp":1780449674000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10846-026-02372-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,15]]},"references-count":82,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,6]]}},"alternative-id":["2372"],"URL":"https:\/\/doi.org\/10.1007\/s10846-026-02372-2","relation":{},"ISSN":["1573-0409"],"issn-type":[{"value":"1573-0409","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,15]]},"assertion":[{"value":"3 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The current study did not involve any human\/animal participants, their data or biological material, and did not require ethics approval.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"59"}}