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However, the modeling of inverse kinematics remains challenging because of its inherent nonlinearities and various external conditions. This work proposes an online learning control framework with a data cache pool utilizing a constant\u2010curvature model inspired neural network (CCMINN) model to obtain the inverse kinematics model of tendon\u2010actuated continuum manipulators. The CCMINN model is a kind of geometric relationships\u2010inspired neural network, which is inspired by the geometric relationships within the constant\u2010curvature model. This model improves the ability of traditional fully connected neural network models on high convergence speed and precision through its constant\u2010curvature inspiration layers. These layers embed geometry insights into the neural network structure rather than loss functions like physics\u2010informed neural networks. The online learning framework enables CCMINN to maintain high control accuracy in a variety of external load scenarios. Experiments show average tracking errors of 1.4\u2009mm, 1.38\u2009mm, and 1.48\u2009mm (0.7%, 0.64%, and 0.74% of the continuum manipulator length) in the free space, under constant and variable loading conditions, respectively. The results show that combining the fast\u2010converging CCMINN with an online learning control framework enables high\u2010precision and robust positioning control of continuum manipulators under various external payloads.<\/jats:p>","DOI":"10.1002\/aisy.202401027","type":"journal-article","created":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T07:25:27Z","timestamp":1756797927000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A High\u2010Precision and Robust Geometric Relationships\u2010Inspired Neural Network for the Inverse Kinematic Modeling of the Tendon\u2010Actuated Continuum Manipulator"],"prefix":"10.1002","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-8135-8614","authenticated-orcid":false,"given":"Jinyu","family":"Duan","sequence":"first","affiliation":[{"name":"Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education School of Mechanical Engineering Tianjin University  Tianjin 300072 China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4860-8619","authenticated-orcid":false,"given":"Jianxiong","family":"Hao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education School of Mechanical Engineering Tianjin University  Tianjin 300072 China"}]},{"given":"Pengyu","family":"Du","sequence":"additional","affiliation":[{"name":"Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education School of Mechanical Engineering Tianjin University  Tianjin 300072 China"}]},{"given":"Bo","family":"Zhang","sequence":"additional","affiliation":[{"name":"Future Robotics Organization Waseda University  1 Chome\u2010104 Totsukamachi Shinjuku City 169\u20108050 Tokyo Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0204-3867","authenticated-orcid":false,"given":"Zhiqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electronic and Electrical Engineering University of Leeds  Leeds LS2 9JT UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9065-9057","authenticated-orcid":false,"given":"Chaoyang","family":"Shi","sequence":"additional","affiliation":[{"name":"Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education School of Mechanical Engineering Tianjin University  Tianjin 300072 China"}]}],"member":"311","published-online":{"date-parts":[[2025,9,2]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1177\/0278364910368147"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1126\/scirobotics.aah3690"},{"key":"e_1_2_8_4_1","volume":"2022","author":"Zhang J.","year":"2022","journal-title":"Cyborg Bionic Syst"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tibtech.2013.03.002"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/aisy.202200367"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1002\/aisy.202000128"},{"key":"e_1_2_8_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/TRO.2015.2489500"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1089\/soro.2019.0083"},{"key":"e_1_2_8_10_1","doi-asserted-by":"crossref","unstructured":"M. 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