基于超多重对偶四元数的刚体运动射流的任意阶埃尔米特插值
Arbitrary-Order Hermite Interpolation of Rigid-Motion Jets via Hyper-Multidual Quaternions
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中文总结 AI 辅助
该研究提出基于超多重对偶四元数的任意阶刚体运动射流埃尔米特插值方法，解决了直接螺杆线性插值的非完整性问题，可恢复高阶加速度场并通过了SE(3)相关测试。
中文摘要 AI 辅助
我们研究由单位对偶四元数表示的有限阶刚体运动射流的双边插值问题。n阶多重对偶（MD）代数是截断多项式代数ℝ[ε]/(ε^(n+1))；超多重对偶（HMD）四元数是系数取自该代数的对偶四元数。时域HMD变换编码位姿及其导数，而一般的HMD曲线不一定是其位姿投影的时域射流，我们将这一要求称为“完整性”。我们证明时域变换及其相对描述符是酉的，并推导递归系数约束，以及在可容许对数图中的局部可实现性逆命题。随后，我们将螺杆线性插值（ScLERP）代数地扩展到单位HMD四元数。尽管该方法匹配完整端点变换，但直接的HMD-ScLERP对于任意端点射流通常是非完整的。我们给出系数准则以及明确的端点和一阶内部接触缺陷。通过将端点变换映射到对数对偶四元数坐标、应用匹配至n阶导数的2n+1次埃尔米特多项式、再通过指数映射提升，可得到一种完整替代方案。HMD算术还可在不显式计算dexp导数的情况下恢复高阶刚体运动加速度场。针对旋转和完整SE(3)进行了至二阶的测试，额外的三阶多项式检验复现了所述缺陷和端点射流。
英文摘要
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.
发表机构
- Technical University of Iasi（雅西技术大学）
- Center for Industrial Robots Simulation and Testing（工业机器人仿真与测试中心）
- Technical University of Cluj-Napoca（克卢日-纳波卡技术大学）
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