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Arbitrary-Order Hermite Interpolation of Rigid-Motion Jets via Hyper-Multidual Quaternions

Daniel Condurache

arXiv:2608.27000v1cs.RO

TL;DR

The paper addresses how to interpolate finite-order rigid-motion jets while preserving the distinction between generic HMD curves and temporal differential transforms. It extends ScLERP algebraically, characterizes its generic non-holonomicity, and constructs a holonomic alternative by Hermite interpolation in logarithmic coordinates followed by exponentiation. Tests through second order and an additional third-order polynomial check reproduce the stated endpoint and contact behavior.

  • Problem

    Finite-order rigid-motion jets require interpolation that distinguishes algebraic HMD group curves from temporal differential transforms of pose trajectories.

  • Method

    The paper combines HMD unit and relative-descriptor constraints with logarithmic-coordinate Hermite interpolation and exponential lifting, using HMD arithmetic for higher-order fields.

  • Results

    Direct HMD–ScLERP is generically non-holonomic, whereas the HMD–Hermite construction matches bilateral finite-order endpoint jets and restores holonomicity.

  • Takeaways & Limitations

    The holonomic construction provides a rigid-motion trajectory interpolant for prescribed endpoint pose, twist, acceleration, and higher-order jet data.

  • Takeaways & Limitations

    The construction is local to an admissible logarithm chart, requires segment splitting at the rotation cut locus, and has Θ(n^2) evaluation cost with increasing multidual-memory demands.

Abstract

from arXiv · show

We study bilateral interpolation of finite-order rigid-motion jets represented by unit dual quaternions. An order-$n$ multidual (MD) algebra is the truncated polynomial algebra $\mathbb{R}[\varepsilon]/(\varepsilon^{n+1})$; hyper-multidual (HMD) quaternions are dual quaternions with coefficients in this algebra. Temporal HMD transforms encode a pose and its derivatives, whereas a generic HMD curve need not be the temporal jet of its pose projection; we call this requirement holonomicity. We show that a temporal transform and its relative descriptor are unitary and derive recursive coefficient constraints, together with a local realizability converse in an admissible logarithm chart. We then extend screw linear interpolation (ScLERP) algebraically to unit HMD quaternions. Although it matches complete endpoint transforms, direct HMD--ScLERP is generically non-holonomic for arbitrary endpoint jets. We give a coefficient criterion and explicit endpoint and first-order interior contact defects. A holonomic alternative is obtained by mapping endpoint transforms to logarithmic dual-quaternion coordinates, applying the degree-$(2n+1)$ Hermite polynomial that matches derivatives through order $n$, and lifting by the exponential. HMD arithmetic also recovers higher-order rigid-motion acceleration fields without explicit differentiation of $\mathrm{dexp}$. Rotation and full $\mathrm{SE}(3)$ tests through second order, with an additional third-order polynomial check, reproduce the stated defects and endpoint jets.

1 Introduction

This paper distinguishes algebraic interpolation of unit HMD quaternions from kinematically valid interpolation of rigid-motion jets. It extends ScLERP to HMD quaternions, characterizes its holonomicity limits, and introduces a Hermite-logarithmic alternative.

  • Problem: Generic HMD curves need not be temporal differential transforms of their pose projections, creating a holonomicity condition beyond group membership.Confusing these levels would overstate what endpoint interpolation guarantees.
  • Contributions: The paper extends ScLERP directly to the Lie group of unit HMD quaternions and states precisely what is interpolated for differential-transform endpoints.These contributions separate algebraic endpoint matching from kinematic interpretation.
  • Contributions: Unit HMD differential transforms and their relative descriptors yield explicit coefficient constraints, including recursive restrictions on higher-order terms.The constraints are derived from the unit relative HMD descriptor.
  • Contributions: A necessary-and-sufficient coefficient criterion, endpoint defect formulas, and a generic non-holonomicity theorem identify when direct HMD–ScLERP fails to represent a rigid-motion history.The paper also distinguishes this obstruction from the algebraic existence of the interpolant.
  • Contributions: HMD arithmetic recovers higher-order acceleration fields from ˘qq∗ without explicit Lie-bracket expansion in the computational construction.The resulting approach handles higher-order kinematic information algebraically.
  • Contributions: The HMD–Hermite construction uses a degree-(2n+1) polynomial in logarithmic coordinates, then applies the exponential to match arbitrary compatible endpoint jets.HMD automatic differentiation supplies endpoint coordinate jets without explicit dexp differentiation.

2 Multidual differential algebra and hyper-multidual representations of rigid motion

The paper builds finite-order multidual and hyper-multidual algebras for encoding derivatives, then extends these operations to rigid-motion tensors and unit dual quaternions. It establishes algebraic preservation, unitary constraints, local realizability conditions, and the distinction between valid temporal jets and generic HMD curves.

  • 2.1 The multidual algebra: The order-n multidual algebra is a finite commutative algebra with nilpotent generator ε, truncated multiplication, and invertibility determined by the zeroth coefficient.Its nilpotent ideal has quotient isomorphic to R, making the algebra local.
  • 2.2 The multidual differential transform: Multidual prolongation preserves compatible functional identities through finite Taylor and Cauchy-product formulas.For temporal transforms, the result is the truncated Taylor shift f(t + ε), with factorials encoding derivatives and the Leibniz rule.
  • 2.3 Dualization of the multidual algebra: Hyper-multidualization adds a commuting dualization generator ε0, whose role is distinct from ε, which records derivatives through order n.The MD transform extends coefficientwise to dual-valued curves, preserving the corresponding dual functional identities.
  • 2.4 HMD tensors and the orthogonal HMD group: HMD tensors extend tensor operations over the multidual coefficient ring, and differential transforms of SO3 curves preserve orthogonality and determinant-one structure.The resulting coefficients encode velocity, acceleration, jerk, and all retained higher-order fields.
  • 2.5 HMD quaternions, unit quaternions, and the tensor covering: An admissible logarithm chart fixes a smooth single branch away from the rotation cut locus, enabling local realization of compatible unit HMD jets as temporal transforms.Unitarity constrains endpoint coefficients but does not by itself guarantee interior holonomicity for a generic HMD curve.
  • 2.5 HMD quaternions, unit quaternions, and the tensor covering: Temporal and relative HMD quaternion descriptors are unitary, with higher coefficient constraints recursively determined by lower-order derivatives.The quaternionic and tensorial descriptors encode the same hierarchy of higher-order kinematic fields.

3 HMD–ScLERP

HMD–ScLERP extends screw linear interpolation to unit HMD quaternions and exactly matches prescribed endpoint transforms. However, the notation does not by itself establish temporal holonomicity.

  • Definition and group interpolation: Algebraic HMD–ScLERP is defined for unit HMD quaternions whose relative element lies in a common admissible logarithm chart.The construction exponentiates the relative logarithm to form a one-parameter subgroup.
  • Definition and group interpolation: The interpolated curve remains in the unit HMD group and satisfies the endpoint interpolation identities.Left equivariance follows because simultaneous left multiplication leaves the relative element unchanged.
  • Endpoint matching: Direct HMD–ScLERP exactly interpolates both complete endpoint differential transforms and all endpoint coefficients they encode.This endpoint result is an equality in the HMD group.
  • Endpoint matching: The interpolant is deliberately denoted as a generic HMD curve until temporal holonomicity has been established.Membership in the unit HMD group alone does not identify its coefficients with successive physical-time derivatives.

4 Holonomicity and contact defects

Temporal holonomicity requires the HMD coefficients to equal the successive time derivatives of the pose projection, not merely to satisfy endpoint or group constraints. The section characterizes these conditions and shows that direct HMD–ScLERP is generically non-holonomic, with a constant-screw exception.

  • Holonomicity criterion: Holonomicity is defined by equality between the HMD curve and the finite temporal differential transform of its pose projection.The coefficient formulation is equivalent to vanishing temporal contact defects at every order.
  • Holonomicity criterion: A curve is temporally holonomic if and only if every contact defect C_k vanishes identically for k = 1, …, n.The criterion is necessary and sufficient because equality of HMD quaternions is coefficientwise equality.
  • Direct HMD–ScLERP defects: Direct HMD–ScLERP is not holonomic for generic endpoint jets, despite matching complete endpoint transforms.Its endpoint defects impose a proper subset of possible endpoint velocities as the first-order compatibility condition.
  • Special case: Direct HMD–ScLERP is temporally holonomic when both endpoint transforms come from the same constant-screw motion.In the commuting subalgebra generated by the screw, normalized time shifts the logarithmic parameter affinely.
  • Direct HMD–ScLERP defects: The first-order interior contact defect is expressed through the Fréchet differentials of Log and exp and vanishes at the endpoints under the stated reductions.The logarithmic perturbation can instead be obtained as the ε coefficient of a finite HMD logarithm.
  • Exponential differential: The inverse of the exponential differential has a closed pure-dual-quaternion form on the rotation chart 0 < ||ξ_r|| < π.The corresponding 8×6 system is consistent for tangent perturbations, while chart behavior becomes singular as θ_r approaches π.

5 Holonomic HMD–Hermite–ScLERP

The holonomic HMD–Hermite–ScLERP construction interpolates endpoint logarithmic jet data with a degree-(2n+1) Hermite polynomial before applying the exponential. Under chart and compatibility assumptions, it matches pose derivatives through order n at both endpoints.

  • Construction: The construction maps endpoint transforms to logarithmic dual-quaternion coordinates, interpolates those coordinates with a Hermite polynomial, and then applies the exponential.HMD arithmetic extracts all endpoint coordinate jets without explicit differentiation of dexp.
  • Construction: The unique degree-at-most-(2n+1) polynomial matches derivatives through order n at both normalized-time endpoints.Existence and uniqueness follow componentwise from scalar Hermite interpolation in the finite-dimensional dual-quaternion Lie algebra.
  • Theorem and assumptions: Theorem 5.2 assumes a common admissible logarithm chart and compatible order-n endpoint differential transforms.Under these conditions, the lifted HMD curve is holonomic and satisfies bilateral jet interpolation.
  • Theorem and assumptions: The resulting interpolant matches pose and all prescribed derivatives through order n at both endpoints.The proof uses equality of coordinate jets and preservation of finite-order jet equality under the analytic exponential map.
  • Properties and limitations: The construction is left-equivariant because simultaneous left multiplication leaves the relative logarithmic data unchanged.Its costs include dependence on the selected logarithm center, lack of general endpoint-reversal symmetry, and non-geodesic behavior.
  • Low-order instances: For first-order data, the cubic Hermite interpolant matches both endpoint transforms and their first-order differential transforms.Higher-order acceleration data are incorporated through the corresponding quintic Hermite construction.

6 Recovery of higher-order acceleration fields

HMD arithmetic packages velocity, acceleration, jerk, and higher-order rigid-motion fields in a single finite algebraic object. It retains noncommutative terms automatically, without explicit differentiation of dexp or expanded Lie-bracket recurrences.

  • Field interpretation: For a genuine HMD differential transform, the constructed HMD field descriptor represents the interpolated rigid motion’s higher-order acceleration fields.Without holonomicity, the analogous HMD product remains defined but cannot automatically be identified with the pose motion’s acceleration fields.
  • Field extraction: Equations (87)–(90) identify velocity, acceleration, jerk, and higher-order acceleration fields within one HMD object.The coefficient convention extracts the kth field using k![ε^k].
  • Computational advantage: HMD quaternion multiplication automatically retains the noncommutative conversion terms needed for these fields.This avoids explicit dexp derivatives and expanded Lie-bracket recurrences in the computational pipeline.

7 Reduction properties and numerical validation

The constructions reduce to ordinary ScLERP and SLERP under algebra reductions, while numerical tests validate non-holonomicity of direct HMD–ScLERP and holonomic endpoint matching of HMD–Hermite.

  • Reduction properties: Removing the multidual unit ε yields ordinary dual-quaternion ScLERP, and removing the dual unit ε0 further yields quaternion SLERP.
  • Numerical validation: The rotation-only test produces strictly positive interior contact defects for noncommuting endpoint directions, although direct HMD–ScLERP matches the endpoint defect formulas.The defect is attributed to the noncommutation of i and j with the k generator.
  • Numerical validation: The cubic HMD–Hermite curve reproduces the first-order endpoint velocities to 1.5 × 10^-14 and 7.9 × 10^-10, with identically zero interior contact defect.Its zero-defect property follows from the curve being a differential transform, not merely from numerical evaluation.
  • Numerical validation: The screw-motion test also exhibits endpoint defects and nonsymmetric sampled values because its endpoint velocities lack the rotation-only test’s discrete symmetry.
  • Numerical validation: The quintic construction is evaluated against first- and second-derivative targets using five-point centered formulas with h = 2 × 10^-4, with residuals reported in Table 3.
  • Higher-order validation: A degree-7 order-n = 3 check gives a maximum double-precision residual of 1.44 × 10^-12 for the eight endpoint identities.The experiment checks the polynomial implementation at n = 3; arbitrary finite-order extension follows from the theorem.
  • Numerical conditioning: The monomial basis becomes progressively less accurate as n increases: its condition number reaches 5.5·10^8 at n = 5, with residual approximately 1.9×10^-9.Adequacy depends on data scaling and tolerance; alternative bases are suggested but not quantitatively assessed.

8 Discussion

The discussion separates algebraic HMD–ScLERP from holonomic trajectory interpolation, relates HMD prolongation to invariant spline constructions, and identifies local scope and extension boundaries.

  • Interpretation: Direct HMD–ScLERP interpolates in the unit-HMD group but is generically non-holonomic, whereas HMD–Hermite interpolates logarithmic endpoint jets before lifting them back.
  • Invariant prolongation: Temporal prolongation of a C^n rotation spline remains orthogonal and equivariant under constant left and right multiplications.
  • Invariant prolongation: This equivariance is weaker than full geometric bi-invariance, which is not claimed for the construction.
  • Tensorial scope: Direct interpolation of arbitrary HMD tensor knots remains unsupported without compatibility with a common temporal source; the original C^2 spline also does not supply immediate jets for n ≥ 3.
  • Applications: HMD–Hermite targets prescribed compatible endpoint jets at arbitrary finite order for applications involving pose, twist, acceleration, jerk, and snap.
  • Limitations: The construction is local because logarithm charts exclude the rotation cut locus, Lie-algebra interpolation does not generally preserve screw invariants, and evaluation costs Θ(n^2).A logarithmic-coordinate Hermite remainder is O(T^(2n+2)); an exponential-lift estimate is not proved.
  • Extensions: Multi-segment and constrained-subgroup extensions preserve finite HMD-product-algebra computation without iterated Lie brackets or Bernoulli series.

9 Conclusion

The paper distinguishes exact group interpolation from temporal holonomicity, characterizes unit HMD differential transforms, and constructs a holonomic Hermite alternative from logarithmic endpoint jets.

  • HMD–ScLERP extends algebraically to unit HMD quaternions, but direct interpolation of arbitrary endpoint transforms is generically non-holonomic.
  • Unit HMD differential transforms satisfy explicit binomial coefficient constraints, while the relative descriptor is unitary and recursively determines higher-order relative fields.
  • The holonomic HMD–Hermite–ScLERP interpolates HMD logarithmic endpoint jets and lifts the resulting Hermite polynomial back to the group.
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