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Lie-Algebraic Bell Recurrences for Arbitrary-Order Twist Jets and Parallel-Mechanism Closure

Daniel Condurache

arXiv:2609.10748v1cs.RO

TL;DR

The paper addresses arbitrary-order kinematics across serial chains, parallel closure, and rigid-platform point fields within one dual screw framework. It combines native cylindrical-joint Bell derivatives with order-preserving propagation and triangular closure, then validates the resulting fields through snap with residuals below 10^-12 under regular active–passive partitions.

  • Problem

    Arbitrary-order rigid-body kinematics must consistently connect serial propagation, differentiated parallel closure, and material-point velocity, acceleration, jerk, and snap fields.

  • Method

    The paper uses native cylindrical blocks, commutative Bell polynomials for fixed axes, order-preserving chain products, covariant twist-jet representations, and an arbitrary-order triangular active–passive closure recurrence.

  • Results

    Independent rigid-motion differentiation, affine-field evaluation, and differentiated branch closures agree through fourth order with residuals below 10^-12 in corresponding SI units.

  • Takeaways & Limitations

    The construction supplies a unified purely kinematic route from serial and parallel twist jets to exact velocity, acceleration, jerk, and snap fields.

  • Takeaways & Limitations

    The formulation assumes a regular selected active–passive partition and does not resolve the kinematic continuation at passive-Jacobian rank loss.

Abstract

from arXiv · show

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.

1 Introduction

The paper builds an arbitrary-order kinematic pipeline from native cylindrical-joint serial propagation through parallel closure to affine platform fields. It combines Bell-polynomial joint derivatives, order-preserving chain products, triangular active–passive recurrences, and complementary validations.

  • 1 Introduction: The construction links ordered serial twist propagation, parallel-mechanism closure, and affine velocity, acceleration, jerk, and snap fields.The platform twist jet is converted algebraically into point-independent affine invariants.
  • 1 Introduction: A native cylindrical joint retains independent rotational and translational coordinates, with revolute and prismatic joints recovered as restrictions.The joint remains one physical block rather than being split into consecutive joints.
  • 1 Introduction: Joint-level Bell polynomials organize derivatives, while noncommuting joint factors remain in physical chain order.Initial-frame prefix and terminal-resolved representations provide covariant branch quantities for common platform closure.
  • 1 Introduction: At each derivative order, the highest passive derivative enters linearly, so one passive-Jacobian factorization can be reused at a regular configuration.The recurrence’s right-hand side depends on prescribed active data and lower-order jets; rank loss removes local uniqueness guarantees.
  • 1 Introduction: Three complementary tests cover serial propagation, active–passive closure, and independently reconstructed spatial platform fields.The 3C chain tests ordered propagation, the RR + RRR wrist tests closure, and the Hunt-type 6-RUS mechanism tests general spatial fields.

2 Condurache’s orthogonal-dual-tensor convention for an mC chain

The paper adopts Condurache’s dual-vector and orthogonal-dual-tensor convention for an ordered chain of native cylindrical joints. Geometric twist derivatives are distinguished from ordinary derivatives of components already resolved in a moving frame.

  • 2 Condurache’s orthogonal-dual-tensor convention for an mC chain: The mechanism is modeled as an ordered chain of bodies connected by general cylindrical joints, with each cylindrical joint represented as one physical block.No cylindrical joint is replaced by consecutive revolute and prismatic joints.
  • 2 Condurache’s orthogonal-dual-tensor convention for an mC chain: A cylindrical joint uses a dual angle with independent rotational and translational components; revolute, prismatic, and helical pairs are restrictions of the same formula.The adopted dual axis combines a unit direction with its line moment.
  • 2 Condurache’s orthogonal-dual-tensor convention for an mC chain: The terminal displacement and transported axes are expressed through ordered relative factors and initial- or terminal-frame resolutions of one geometric dual line vector.The two axis families are covariant representations rather than differentiated coordinate curves.
  • 2 Condurache’s orthogonal-dual-tensor convention for an mC chain: The derivative convention separates geometric twist derivatives from ordinary derivatives of already resolved moving-frame components.This prevents the moving-frame derivative error represented by (mW^m)^(n) ≠ ^mW⟨n⟩.

3 Commutative Bell polynomials for a fixed dual axis

For a fixed dual axis, derivatives of the exponential factor are organized by ordinary commutative Bell polynomials because all generator derivatives are scalar multiples of one constant tensor. Noncommutativity is retained only across different joint factors.

  • 3 Commutative Bell polynomials for a fixed dual axis: The fixed-axis generator derivatives commute because they are scalar multiples of the same constant cross-product tensor.Ordered or noncommutative Bell families are unnecessary for one joint.
  • 3 Commutative Bell polynomials for a fixed dual axis: A fixed-axis joint factor has derivatives expressed by ordinary partial and complete exponential Bell polynomials.The fixed-axis derivative reduces to a tensor basis generated by the joint’s unit dual axis.
  • 3 Commutative Bell polynomials for a fixed dual axis: Different joint factors remain in physical chain order and are combined with the multinomial Leibniz rule.Thus commutative Bell polynomials operate at joint level while the chain product remains order preserving.
  • 3 Commutative Bell polynomials for a fixed dual axis: The resulting factor derivatives support arbitrary-order serial twist formulas using ordinary Bell polynomials and order-preserving products.The formulation avoids replacing the chain-level ordering with a commutative product.

4 Higher-order kinematics of the mC chain in the initial frame

The initial-frame construction differentiates an ordered native mC chain by combining fixed-axis Bell factors with order-preserving Leibniz products. It yields an explicit arbitrary-order twist jet while remaining confined to the fixed initial frame.

  • Ordered chain differentiation: The mC chain preserves physical joint order, using commutative Bell factors only within each fixed-axis cylindrical joint.The multinomial coefficient distributes derivatives among relative tensors without permuting factors.
  • Ordered chain differentiation: The forward recurrence is exactly equivalent to the closed multinomial expression because both apply Leibniz differentiation to the ordered product.The recurrence follows by differentiating 0Rk−1Qk.
  • Resolved axes: All time dependence of each initial-frame resolved joint axis is carried by the ordered prefix preceding that joint.The joint axis itself is fixed in its native joint frame.
  • Twist-jet formula: The resulting arbitrary-order twist jet is explicit in Bell polynomials, joint dual-angle derivatives, constant axes, and order-preserving relative-tensor products.This is the complete higher-order twist-jet formula for the mC chain.
  • Scope: The initial-frame result does not assert term-by-term equivalence with Condurache’s operator-polynomial organization without a separate reindexing proof.The formula is used in its autonomous Leibniz form.
  • Scope: The construction does not differentiate moving-frame components or introduce a space–body conversion; it remains fixed to the initial frame.Its ingredients are the mC product, fixed dual axes, joint Bell factors, and ordinary Leibniz differentiation.

5 Higher-order kinematics of the mC chain in the terminal frame

The terminal-frame formulation is obtained covariantly from the initial-frame twist jet, resolving geometric derivatives in the terminal body without differentiating moving-frame component arrays. It preserves Bell coefficients and the physical order of distinct joints.

  • Terminal resolution: The terminal formulation resolves the geometric derivative of the terminal twist in Cm rather than differentiating time-dependent components in the moving frame.It is the terminal-frame counterpart of the initial-frame description.
  • Covariance and ordering: Changing the resolution frame changes the generator but not the ordinary commutative Bell coefficients, while distinct-joint products retain chain order.Commutativity remains restricted to factors belonging to one joint.
  • Covariance proof: A covariance lemma establishes the terminal expression term by term for every derivative order.The proof inserts proper orthogonal transformations without commuting factors from different joints.
  • Evaluation: The terminal-resolved formula evaluates joint contributions at the same instant as the resolved axes and does not differentiate moving-frame coordinates.The square-index derivatives collect preceding joint contributions in physical order.
  • Geometric differentiation: The first five members of the geometric jet distinguish geometric differentiation from ordinary coordinate differentiation.The section therefore avoids descending suffix states, differentiated space–body identities, and extra transport recurrences.
  • Validation: A generic noncoplanar 3C chain with nonzero rotational and translational coordinates tests the ordered construction through fourth order.Independent product differentiation gives discrepancies below 10^-9, while initial/terminal covariance closes below 10^-15.

6 Dual model and closure equations of a parallel mechanism

The parallel-mechanism construction evaluates each branch as an open native-cylindrical serial chain, assembles branch quantities in a common platform frame, and imposes real active–passive closure equations. A spherical-wrist example reconstructs the platform twist with residuals below 10^-12 using unrounded values.

  • Branch model: Each branch is modeled as an open serial chain, while the common platform supplies the closure equations linking branch predictions.Base and platform attachment tensors keep branch terminal bodies distinct from the platform frame.
  • Closure equations: Branch platform displacements and twists are compared in a common resolving frame, producing stacked closure equations relative to a reference branch.The reference branch is an algebraic choice and has no distinguished physical role.
  • Realification and partition: The complete dual system is realified by replacing each dual column with a real block, then partitioned into prescribed active and dependent passive coordinates.The complete stacked comparison system and the independent local system share the same admissible tangent space.
  • Regularity: At a regular configuration, a square nonsingular passive block uniquely determines passive rates from prescribed active rates.The inverse formula is reserved for the square independent system.
  • Regularity boundary: The selected active–passive partition is local: passive-block rank loss destroys local uniqueness, while complete-system rank loss indicates dependent closure equations.These rank tests depend on the chosen actuation partition and must not be conflated.
  • Spherical-wrist example: The RR + RRR spherical wrist has five concurrent revolute axes at O, two actuated branch joints, and zero translational twist at the common center.Its two branches are RR and RRR, with the first joint of each branch actuated.
  • Spherical-wrist validation: Residuals below 10^-12 are obtained for the unrounded spherical-wrist closure solve, while the reconstructed platform twist agrees from either branch.The printed six-decimal values instead give a residual below 10^-6.

7 Bell recurrence for arbitrary-order parallel closure

The paper extends parallel-mechanism closure to arbitrary derivative order using Bell-organized repeated Leibniz differentiation. At a regular configuration, the highest passive derivative enters linearly, so one passive-Jacobian factorization serves every order while lower-order terms populate changing right-hand sides.

  • Branchwise recurrence: Repeated Leibniz differentiation isolates the sole highest-order derivative linearly because Bell-polynomial residuals contain only derivatives through the current order.This yields an ascending-order recurrence using prescribed active data and previously computed jets.
  • Arbitrary-order closure: The triangular Bell closure theorem makes branch twist agreement equivalent to the stacked active–passive closure equations at every derivative order.At order zero it reduces to the first-order closure, while higher-order nonlinear terms remain on the known right-hand side.
  • Branchwise recurrence: Prefix and terminal-resolved branch quantities are assembled in a common platform frame, preserving physical chain order while enabling closure across branches.The wrist continuation uses the same numerical passive system at orders two through four, with moving-axis derivatives included in Bell residuals.
  • Recursive evaluation: One factorization of the passive Jacobian produces the entire passive jet; only the right-hand side changes with derivative order.The matrices are evaluated once at the current configuration and are not differentiated.
  • Validation: The RR + RRR wrist continuation reconstructs the platform angular jet through fourth derivative order and closes every order to displayed precision.Its independently reproduced passive derivatives and platform angular jet provide a numerical check of the continuation.
  • Recursive evaluation: The recurrence requires a regular active–passive partition; Bell polynomials organize nonlinear terms but do not remove singularities.A redundant rectangular system additionally requires pseudoinverse range compatibility at every order.

8 Affine invariants and higher-order vector fields of the platform

The platform twist jet is mapped algebraically to point-independent affine invariants and then to velocity, acceleration, jerk, and snap fields. An independently reconstructed Hunt-type 6-RUS example verifies the twist-to-field stage and fourth-order consistency without an additional kinematic solve.

  • Exact twist-to-field map: The construction assumes a fixed frame for ordinary derivatives and applies the exact map without reconstructing pose or its derivatives.Body-resolved twist data must first be transported to the fixed frame.
  • Exact twist-to-field map: The operator-polynomial map converts a fixed-frame twist jet into point-independent vector–tensor invariants and affine fields at any platform point.For order n, the field has the form a[n]_ρ = a_n + Φ_nρ, with orders one through four corresponding to velocity through snap.
  • Exact twist-to-field map: The twist jet through order three is sufficient for complete velocity, acceleration, jerk, and snap fields, with no additional affine-stage kinematic solve.Conditioning loss therefore belongs to the preceding closure stage rather than the twist-to-field map.
  • 6-RUS validation: The 6-RUS platform exhibits nonzero vector invariants at every reported order, supporting its use for general spatial platform motion.The configuration uses six active revolutes and a spatial moving platform rather than the spherical wrist’s didactic closure setting.
  • 6-RUS validation: The 6-RUS validation independently differentiates exact rod constraints and compares direct rigid-motion fields with the operator-polynomial fields through fourth order.The route does not reuse the branchwise Bell recurrence and does not reconstruct passive universal or spherical-joint coordinates.
  • 6-RUS validation: All six closure equations and rotational orthogonality conditions vanish through order four with residuals below 10^-12 in corresponding SI units.These are internal floating-point consistency checks rather than comparisons against rounded published outputs.

9 Conclusions

The paper unifies ordered serial propagation, triangular active–passive closure, and exact rigid-platform field reconstruction in an arbitrary-order dual-screw kinematic route. Complementary mechanism checks agree through snap with residuals below 10^-12 within the stated regular kinematic domain.

  • Framework: A continuous construction separates serial branch propagation, active–passive closure, and affine-field reconstruction for parallel mechanisms.Keeping these stages distinct makes the algebra and numerical checks explicit.
  • Serial construction: Native cylindrical joints encode coaxial rotation and translation with one dual scalar, while ordered products preserve inter-joint noncommutativity.Initial-frame and terminal-resolved covariant forms provide equivalent branch twist-jet descriptions for common platform closure.
  • Parallel closure: At every derivative order, the highest passive derivative enters linearly, allowing one passive-Jacobian factorization to be reused at a fixed regular configuration.Lower-order jets supply the nonlinear right-hand-side terms; rank loss is not regularized by the recurrence.
  • Platform fields: The closed platform twist jet maps exactly to point-independent pairs that generate velocity, acceleration, jerk, and snap at every material point.No additional mechanism solve is required after closure; active–passive conditioning is inherited from that closure problem.
  • Validation: Independent rigid-motion differentiation, affine-field evaluation, and six branch closures agreed through fourth order with residuals below 10^-12 in corresponding SI units.The checks used a generic 3C chain, an RR + RRR spherical wrist, and a Hunt-type 6-RUS configuration with independent platform-jet reconstruction.
  • Scope: The framework is purely kinematic and applies to fixed-topology mechanisms with a regular active–passive partition, not singular continuation, branch switching, dynamics, compliance, or contact.The stated extensions include rank-deficient closure, high-order conditioning and error propagation, and integration with higher-order dynamics or trajectory generation.
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