发表机构
Gheorghe Asachi Technical University of Iaşi（雅西格奥尔基·阿萨基技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出任意阶运动学构造，用贝尔多项式统一串联传播与并联闭式解，实现平台射流至仿射不变量映射，并经多机构验证至四阶精度。
AI 中文摘要
本文发展了一种任意阶运动学构造，在单一对偶螺旋框架内将串联传播、并联机构闭式解和刚体平台点场联系起来。圆柱关节被保留为一个原生物理块，转动关节和移动关节作为其特例获得。对于每个固定关节轴，普通贝尔多项式组织指数因子的导数；沿链，非交换因子保持其物理顺序。初始坐标系前缀和末端解析协变公式产生串联扭转射流的等价表示。对于并联机构，重复莱布尼茨微分，关节级导数由贝尔多项式组织，产生任意阶三角主动-被动闭式递推：在正则位形下，每个导数阶求解相同的被动雅可比矩阵，而右侧仅包含指定的主动数据和低阶射流。所得平台扭转射流精确映射到速度、加速度、加加速度和急动度场的点无关仿射不变量。验证刻意互补：具有非共面轴和非零旋转及平移圆柱坐标的通用3C链测试有序串联传播，RR+RRR球腕测试主动-被动闭式解，具有六个主动转动关节的Hunt型6-RUS机构测试独立重建的平台射流及其仿射场。刚体运动的独立微分、仿射场评估和微分支链闭式解均一致至四阶，残差在相应国际单位制下低于10^-12。该公式纯运动学，适用于所选主动-被动划分正则的位形。
英文摘要
This paper develops an arbitrary-order kinematic construction that links serial propagation, parallel-mechanism closure, and rigid-platform point fields within one dual screw framework. A cylindrical joint is retained as one native physical block, with revolute and prismatic joints obtained as special cases. For each fixed joint axis, ordinary Bell polynomials organize the derivatives of the exponential factor; across a chain, the noncommuting factors remain in their physical order. Initial-frame prefix and terminal-resolved covariant formulas then produce equivalent representations of the serial twist jet. For a parallel mechanism, repeated Leibniz differentiation, with joint-level derivatives organized by Bell polynomials, yields an arbitrary-order triangular active-passive closure recurrence: the same passive Jacobian is solved at every derivative order at a regular configuration, while the right-hand side contains only prescribed active data and lower-order jets. The resulting platform twist jet is mapped exactly to the point-independent affine invariants of the velocity, acceleration, jerk, and snap fields. The validation is deliberately complementary: a generic 3C chain with noncoplanar axes and nonzero rotational and translational cylindrical coordinates tests ordered serial propagation, an RR+RRR spherical wrist tests active-passive closure, and a Hunt-type 6-RUS mechanism with six active revolute joints tests an independently reconstructed platform jet and its affine fields. Independent differentiation of the rigid motion, evaluation of the affine fields, and the differentiated branch closures all agree through fourth order with residuals below $10^{-12}$ in the corresponding SI units. The formulation is purely kinematic and applies at configurations where the selected active-passive partition is regular.
Comments24 pages, 2 figures. Ancillary files: Python validation scripts (generic 3C chain, RR+RRR spherical wrist, Hunt-type 6-RUS) and numerical results (JSON)
