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[Submitted on 9 Sep 2026] Title:Lie-Algebraic Bell Recurrences for Arbitrary-Order Twist Jets and Parallel-Mechanism Closure View a PDF of the paper titled Lie-Algebraic Bell Recurrences for Arbitrary-Order Twist Jets and Parallel-Mechanism Closure, by Daniel Condurache View PDF HTML (experimental) Abstract:This paper develops an arbitrary-order kinematic construction that links serial propagation, parallel-mechanism closure, and rigid-platform point fields within one dual screw framework. A cylindrical joint is retained as one native physical block, with revolute and prismatic joints obtained as special cases. For each fixed joint axis, ordinary Bell polynomials organize the derivatives of the exponential factor; across a chain, the noncommuting factors remain in their physical order. Initial-frame prefix and terminal-resolved covariant formulas then produce equivalent representations of the serial twist jet. For a parallel mechanism, repeated Leibniz differentiation, with joint-level derivatives organized by Bell polynomials, yields an arbitrary-order triangular active-passive closure recurrence: the same passive Jacobian is solved at every derivative order at a regular configuration, while the right-hand side contains only prescribed active data and lower-order jets. The resulting platform twist jet is mapped exactly to the point-independent affine invariants of the velocity, acceleration, jerk, and snap fields. The validation is deliberately complementary: a generic 3C chain with noncoplanar axes and nonzero rotational and translational cylindrical coordinates tests ordered serial propagation, an RR+RRR spherical wrist tests active-passive closure, and a Hunt-type 6-RUS mechanism with six active revolute joints tests an independently reconstructed platform jet and its affine fields. Independent differentiation of the rigid motion, evaluation of the affine fields, and the differentiated branch closures all agree through fourth order with residuals below $10^{-12}$ in the corresponding SI units. The formulation is purely kinematic and applies at configurations where the selected active-passive partition is regular. Comments: 24 pages, 2 figures. Ancillary files: Python validation scripts (generic 3C chain, RR+RRR spherical wrist, Hunt-type 6-RUS) and numerical results (JSON) Subjects: Robotics (cs.RO) MSC classes: 70B15 (Primary), 53A17, 11B73 (Secondary) Cite as: arXiv:2609.10748 [cs.RO] (or arXiv:2609.10748v1 [cs.RO] for this version) https://doi.org/10.48550/arXiv.2609.10748 arXiv-issued DOI via DataCite (pending registration) Submission history From: Daniel Condurache [view email] [v1] Wed, 9 Sep 2026 18:45:19 UTC (34 KB) Full-text links: Access Paper: View a PDF of the paper titled Lie-Algebraic Bell Recurrences for Arbitrary-Order Twist Jets and Parallel-Mechanism Closure, by Daniel Condurache View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files (details): requirements.txt results_6rus_affine.json validate_6rus_affine.py validate_generic_3c.py validate_wrist.py Current browse context: cs.RO new | recent | 2026-09 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)