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

The relative rotation between the rigid bodies is $$\begin{equation} \mathbf{Q}=\boldsymbol{\Lambda}^{\left(2\right)}\cdot\boldsymbol{\Lambda}^{\left(1\right)T}=\sum_{j}\mathbf{e}_{j}^{\left(2\right)}\otimes\mathbf{e}_{j}^{\left(1\right)}\,.\label{eq:prj-1} \end{equation}$$ We want it to be equal to a rotation by $\boldsymbol{\chi}$ as expressed in the basis $\mathbf{e}_{j}^{\left(1\right)}$ , $$\begin{equation} \mathbf{A}=\exp\left[\boldsymbol{\chi}\right]\,.\label{eq:prj-2} \end{equation}$$ We enforce this constraint by requiring that $$\begin{equation} \mathbf{A}\cdot\mathbf{Q}^{T}=\mathbf{I}\,.\label{eq:prj-3} \end{equation}$$ We may thus evaluate $$\begin{equation} \mathbf{R}=\mathbf{A}\cdot\mathbf{Q}^{T}\,,\label{eq:prj-4} \end{equation}$$ and expect that $\mathbf{R}=\mathbf{I}$ when the constraint is enforced. Note that $\mathbf{R}^{T}\cdot\mathbf{R}=\mathbf{I}$ because of the orthogonality of $\mathbf{Q}$ and $\mathbf{A}$ , i.e., $\mathbf{R}$ is always orthogonal, even when the constraint is not enforced. The axial vector of $\mathbf{R}$ may be denoted by $\boldsymbol{\xi}$ , i.e., $\mathbf{R}=\exp\left[\boldsymbol{\xi}\right]$ , and the constraint is enforced when $\boldsymbol{\xi}=\mathbf{0}$ . Therefore, a moment $\varepsilon_{r}\boldsymbol{\xi}$ needs to be prescribed, $$\begin{equation} \mathbf{m}^{\left(1\right)}=\varepsilon_{r}\boldsymbol{\xi}\,.\label{eq:prj-5} \end{equation}$$