$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

Joint axes are used to define the directions of degrees of freedom in rigid joints, which represent one of the major categories of rigid connectors. The axes are defined with respect to a body-based coordinate system centered at the origin of a joint, given by the orthonormal triad $\left\{ \mathbf{e}_{1}^{\left(i\right)},\mathbf{e}_{2}^{\left(i\right)},\mathbf{e}_{3}^{\left(i\right)}\right\}$ for rigid body $i$ ($i=1,2$ ). In the reference configuration, the bases coincide on both rigid bodies, $\mathbf{e}_{j}^{\left(1\right)}=\mathbf{e}_{j}^{\left(2\right)}\equiv\mathbf{E}_{j}$ ($j=1,2,3$ ), where $\mathbf{E}_{j}$ is the $j-$ th basis vector in the reference configuration. Thus, at any time $t$ , we may evaluate the basis vectors as $$\begin{equation} \mathbf{e}_{j}^{\left(i\right)}=\boldsymbol{\Lambda}^{\left(i\right)}\cdot\mathbf{E}_{j}\label{eq:ja-basis-currnet-time} \end{equation}$$ Using this relation, the linearization of a basis vector along increments in the rigid body motion is given by $$\begin{equation} D\mathbf{e}_{j}^{\left(i\right)}=-\hat{\mathbf{e}}_{j}^{\left(i\right)}\cdot\Delta\boldsymbol{\theta}^{\left(i\right)}\,,\label{eq:ja-basis-vector-linearization} \end{equation}$$ which shows a dependence only on $\Delta\boldsymbol{\theta}^{\left(i\right)}$ .