[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
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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)
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View a PDF of the paper titled Lie-Algebraic Bell Recurrences for Arbitrary-Order Twist Jets and Parallel-Mechanism Closure, by Daniel Condurache
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Ancillary files (details):
requirements.txt
results_6rus_affine.json
validate_6rus_affine.py
validate_generic_3c.py
validate_wrist.py
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