{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T14:56:01Z","timestamp":1774450561968,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T00:00:00Z","timestamp":1774396800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"College Student Innovation and Entrepreneurship Training Program\n\t\t\t\t\t\t\t\t\t              https:\/\/ror.org\/05meqs994","award":["20250100142"],"award-info":[{"award-number":["20250100142"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Rigid-body chain following on piecewise analytic paths is a fundamental subroutine in motion planning and multibody simulation. The problem is nontrivial when only the leader trajectory of the first node is available: enforcing fixed inter-node distances reduces to circle\u2013curve intersection, which is generally multi-valued and becomes particularly challenging near non-smooth junctions. We present a Dichotomy Geometric Path Search (DGPS) framework that converts each constraint into a one-dimensional root-finding task and resolves the branch selection through no-backtracking ordering: at every time step, the admissible solution for the current node is the nearest feasible root in the past relative to its immediately preceding node. DGPS combines backward bracketing with bisection, achieving robust convergence. Compared with the inverse Jacobian method, which maps end-effector velocities to joint velocities via explicit inversion, the proposed approach avoids Jacobian inversion and globally coupled nonlinear solves. We further characterize the local structure of the zero set and establish monotonicity\/uniqueness conditions that justify stable root selection across piecewise junctions. Extensive tests on representative piecewise trajectories (line\u2013arc\u2013line, polylines with corners, piecewise sinusoids, and time reparameterization) show that DGPS enforces distance constraints to near machine precision, produces interpretable speed\/acceleration transients around non-smooth events, and exhibits computational costs consistent with iteration difficulty. The results support DGPS as a general, efficient solver requiring only the prescribed leader trajectory.<\/jats:p>","DOI":"10.3390\/a19040252","type":"journal-article","created":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T13:57:34Z","timestamp":1774447054000},"page":"252","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Rigid-Chain Following and Kinematic Response Analysis on Piecewise Non-Smooth Paths: A DGPS-Based Solution Method"],"prefix":"10.3390","volume":"19","author":[{"given":"Yaxuan","family":"Zhao","sequence":"first","affiliation":[{"name":"National \u201c111\u201d Research Center for Microelectronics and Integrated Circuits, School of Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-6731-7197","authenticated-orcid":false,"given":"Ziheng","family":"Li","sequence":"additional","affiliation":[{"name":"National \u201c111\u201d Research Center for Microelectronics and Integrated Circuits, School of Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9785-8294","authenticated-orcid":false,"given":"Hualu","family":"Liu","sequence":"additional","affiliation":[{"name":"National \u201c111\u201d Research Center for Microelectronics and Integrated Circuits, School of Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Peng, Y., Yang, L., Sun, Y., and Chen, X. (2025). Novel claw-type continuum robots: Design, modeling, and control. Front. Mech. Eng., 20.","DOI":"10.1007\/s11465-025-0832-8"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"031014","DOI":"10.1115\/1.4050097","article-title":"A new extensible continuum manipulator using flexible mechanisms","volume":"13","author":"Liu","year":"2021","journal-title":"J. Mech. Robot."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"104059","DOI":"10.1016\/j.mechmachtheory.2020.104059","article-title":"Higher-order derivatives of rigid body dynamics with application to the dynamic balance of spatial linkages","volume":"155","author":"Herder","year":"2021","journal-title":"Mech. Mach. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"105613","DOI":"10.1016\/j.mechmachtheory.2024.105613","article-title":"Kinematic jerk and jounce for multibody dynamics with joint constraints","volume":"196","author":"Sommer","year":"2024","journal-title":"Mech. Mach. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1002\/msd2.12037","article-title":"Multibody system transfer matrix method: The past, the present, and the future","volume":"2","author":"Rui","year":"2022","journal-title":"Int. J. Mech. Syst. Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.1017\/S0263574721000345","article-title":"Computational efficiency of multi-body systems dynamic models","volume":"39","author":"Antonya","year":"2021","journal-title":"Robotica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1007\/s11071-021-06731-6","article-title":"A general framework for modeling and dynamic simulation of multibody systems using factor graphs","volume":"105","author":"Leanza","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.apm.2022.10.053","article-title":"An innovative joint-space explicit dynamics symbolic computation model for closed-chain mechanisms","volume":"115","author":"Guo","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"105809","DOI":"10.1016\/j.mechmachtheory.2024.105809","article-title":"A new formulation for the dynamics of rigid bodies with unilateral interactions","volume":"203","year":"2024","journal-title":"Mech. Mach. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Guo, H., Zhang, X., and Wang, K. (2024). A Novel Explicit Canonical Dynamic Modeling Method for Multi-Rigid-Body Mechanisms Considering Joint Friction. Aerospace, 11.","DOI":"10.3390\/aerospace11050368"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3788","DOI":"10.1109\/TNNLS.2019.2946290","article-title":"An Improved PSO Algorithm for Smooth Path Planning of Mobile Robots using Continuous High-Degree Bezier Curve","volume":"31","author":"Shen","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"00368504211063072","DOI":"10.1177\/00368504211063072","article-title":"A hierarchical approach for rigid-body dynamics model simplification of a high-speed parallel robot by considering kinematics performance","volume":"104","author":"Ni","year":"2021","journal-title":"Sci. Prog."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1017\/S0263574722001370","article-title":"A novel inverse kinematics for solving repetitive motion planning of 7-DoF SRS manipulator","volume":"41","author":"Zhao","year":"2023","journal-title":"Robotica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11044-021-09786-w","article-title":"Nonsmooth spatial frictional contact dynamics of multibody systems","volume":"53","author":"Wang","year":"2021","journal-title":"Multibody Syst. Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1145\/3450626.3459802","article-title":"Intersection-free Rigid Body Dynamics","volume":"40","author":"Ferguson","year":"2021","journal-title":"ACM Trans. Graph."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1007\/s11071-021-06344-z","article-title":"Nonlinear phenomena of contact in multibody systems dynamics: A review","volume":"104","author":"Corral","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Flores, P., and Lankarani, H.M. (2016). Contact Force Models for Multibody Dynamics, Springer.","DOI":"10.1007\/978-3-319-30897-5"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Godage, I.S., Branson, D.T., Guglielmino, E., Medrano-Cerda, G.A., and Caldwell, D.G. (2011, January 9\u201313). Shape function-based kinematics and dynamics for variable length continuum robotic arms. Proceedings of the 2011 IEEE International Conference on Robotics and Automation, Shanghai, China.","DOI":"10.1109\/ICRA.2011.5979607"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ran, K., Wang, Y., Fang, C., Chai, Q., Dong, X., and Liu, G. (2024). Improved RRT* Path-Planning Algorithm Based on the Clothoid Curve for a Mobile Robot Under Kinematic Constraints. Sensors, 24.","DOI":"10.3390\/s24237812"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2991","DOI":"10.1109\/LCSYS.2024.3520026","article-title":"Generalization of Optimal Geodesic Curvature Constrained Dubins\u2019 Path on Sphere With Free Terminal Orientation","volume":"8","author":"Kumar","year":"2024","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"105959","DOI":"10.1016\/j.mechmachtheory.2025.105959","article-title":"An overview of higher-order kinematics of rigid body and multibody systems with nilpotent algebra","volume":"209","author":"Condurache","year":"2025","journal-title":"Mech. Mach. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Di Gregorio, R., and Cinti, T. (2024). Geometric Constraint Programming (GCP) Implemented Through GeoGebra to Study\/Design Planar Linkages. 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