cs.ROAug 28, 2026

Dual Park-Ravani Interpolation of Rigid Motions: Acceleration-Field Continuity and Holonomic Hermite Repair

Authors: Daniel Condurache

Organizations: “Gheorghe Asachi” Technical University of Ia¸i, Romania

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Abstract

The Park-Ravani construction generates a twice continuously differentiable, frame-invariant spline on SO(3) by exponentiating cubic canonical-coordinate polynomials. We show that the construction transfers, without changing form, to the group of orthogonal dual tensors, a representation of rigid displacements. The transferred recurrence is stated compactly through the dual extension of the right Jacobian of the exponential map and its first Fréchet derivative. This yields interpolation of prescribed rigid poses and continuity of the body dual twist and its first derivative. Using the higher-order rigid-body kinematics of dual spatial twists, we then prove that the resulting curve has a continuous physical acceleration field, not merely a continuous quantity obtained by formally differentiating the dual part of a twist. We also distinguish algebraic dual transfer from temporal differential prolongation: their simultaneous first-order use takes place in a hyper-dual algebra, and interpolation of arbitrary prolonged nodal data need not be holonomic. A noncommuting three-pose example verifies the recurrence, all knot continuity statements, and dimensional covariance under a change from meters to millimeters. We define and analyze the first-order holonomy defect of a generic hyper-dual interpolant, exhibit an exact counterexample, and remove the defect by cubic or quintic Hermite interpolation in dual logarithmic coordinates.

Figures & tables

VerificationResidual
Compact versus expanded recurrence8.33×10−178.33\times 10^{-17}
Dual pose interpolation8.05×10−168.05\times 10^{-16}
Body dual twist continuity1.36×10−161.36\times 10^{-16}
Body dual-twist derivative continuity2.29×10−162.29\times 10^{-16}
Spatial dual twist continuity2.00×10−162.00\times 10^{-16}
Spatial dual-twist derivative continuity2.29×10−162.29\times 10^{-16}
Table 1: Residuals at the internal knot t1t_{1} .
Pointaxa_{x}aya_{y}aza_{z}
p1p_{1}1.336338-1.7924701.773506
p1+0.25e1p_{1}+0.25e_{1}1.284059-1.7544101.928334
p1+0.20e2p_{1}+0.20e_{2}1.256146-1.8409931.788991
p1+0.18e3p_{1}+0.18e_{3}1.267878-1.8440181.728332
Table 2: Physical point accelerations at t1t_{1} in m s−2\mathrm{m\,s^{-2}} .
ResidualInitial endpointFinal endpoint
Cubic pose008.05×10−168.05\times 10^{-16}
Cubic body twist7.14×10−137.14\times 10^{-13}4.04×10−124.04\times 10^{-12}
Quintic pose008.05×10−168.05\times 10^{-16}
Quintic body twist1.59×10−131.59\times 10^{-13}1.73×10−121.73\times 10^{-12}
Quintic body-twist derivative1.08×10−101.08\times 10^{-10}6.32×10−96.32\times 10^{-9}
Table 3: Independent curve-level verification of the Hermite interpolants.
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