Abstract The most general continuous rigid motion can be described by a curve in SE(3) of whose tangent vector the left or right invariant representation is a curve in se(3). The integral operation between them has not been achieved because there is no general solution for a system of the first-order linear differential equations with variable coefficients in mathematics. This paper develops a matrix equation (termed as the fundamental equation) based on the geometric properties of a pair of conjugate axodes equivalent to a curve in se(3) in algebras to achieve this integral operation. Moreover, the first-order and second-order derivations of the fundamental equation are derived. Furthermore, an algebraic method representing the body and spatial velocity twists as vector functions of dimensions and input parameters of mechanisms is founded on the theory of reciprocal screws. After that, this method and the fundamental equation are validated by the numerical examples of Bennett, spherical and planar four-bar linkages. Finally, this work presents a notion of generalized-involute which is a useful tool in the study of gear tooth profile and cam profile.

中文翻译：

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机构运动几何的基本方程：将 se(3) 中的曲线映射到 SE(3) 中的对应曲线

摘要 最一般的连续刚性运动可以用SE(3)中的一条曲线来描述，其切向量的左或右不变表示是se(3)中的一条曲线。它们之间的积分运算尚未实现，因为数学上没有对变系数一阶线性微分方程组的通解。本文基于一对共轭轴的几何性质，开发了一个矩阵方程（称为基本方程），相当于代数中 se(3) 中的一条曲线，以实现这种积分运算。此外，还导出了基本方程的一阶和二阶导数。此外，将身体和空间速度扭曲表示为机构的维度和输入参数的向量函数的代数方法建立在互易螺旋理论基础上。之后，通过贝内特、球面和平面四连杆机构的数值例子验证了该方法和基本方程。最后，这项工作提出了广义渐开线的概念，它是研究齿轮齿廓和凸轮廓线的有用工具。