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This paper investigates the interpolation of finite-order rigid-motion jets using hyper-multidual quaternions, focusing on the challenges of ensuring holonomicity in the interpolation process. The authors derive recursive coefficient constraints and establish a method for extending screw linear interpolation (ScLERP) to unit HMD quaternions, revealing that while direct HMD-ScLERP matches endpoint transforms, it is generally non-holonomic. To achieve holonomic interpolation, they propose a novel approach that utilizes logarithmic dual-quaternion coordinates and Hermite polynomials, effectively recovering higher-order motion fields without explicit differentiation.
Achieving holonomic interpolation for rigid-motion jets could revolutionize how we model complex motion in robotics and computer graphics.
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.
