刚体运动的对偶Park-Ravani插值：加速度场连续性与完整约束埃尔米特修复
Dual Park-Ravani Interpolation of Rigid Motions: Acceleration-Field Continuity and Holonomic Hermite Repair
AI总结：
本文将Park-Ravani刚体运动插值构造推广到正交对偶张量群，证明所得曲线具有连续物理加速度场，区分两种对偶推广的差异，通过超对偶代数分析完整约束缺陷并以埃尔米特插值消除该缺陷。
AI中文摘要：
Park-Ravani构造通过对三次正则坐标多项式取指数，在特殊正交群SO(3)上生成了一个二阶连续可微、与坐标系无关的样条曲线。本文证明该构造可保持形式不变地推广到正交对偶张量群（刚体位移的一种表示），其推广后的递推关系通过指数映射右雅可比的对偶扩展及其一阶弗雷歇导数紧凑表述，可实现给定刚体位姿的插值，并保证刚体对偶扭转及其一阶导数的连续性。利用对偶空间扭转的高阶刚体运动学，本文进一步证明所得曲线具有连续的物理加速度场，而非仅通过形式上对扭转的对偶部分求导得到的连续量。本文还区分了代数对偶推广与时间微分延拓：二者的一阶同时应用发生在超对偶代数中，且任意延拓节点数据的插值未必是完整约束的。一个非交换的三姿态示例验证了该递推关系、所有节点连续性声明，以及从米到毫米的单位变换下的维数协变性。本文定义并分析了一般超对偶插值的一阶完整约束缺陷，给出了一个精确反例，并通过在对偶对数坐标中进行三次或五次埃尔米特插值消除了该缺陷。
英文摘要：
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.
