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

2026-08-27

AI Summary

本文研究了通过超多对偶四元数实现任意阶刚体运动喷流的Hermite插值方法，解决了非完整插值问题。

Abstract

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.

Problem

• Hermite Interpolation
• Rigid-Motion Jets
• Hyper-Multidual Quaternions
• Holonomicity

Innovation

• Hyper-Multidual Quaternions
• Hermite Interpolation
• Holonomicity
• ScLERP
• Rigid-Motion Jets