$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$

The Rigid section is used to define all rigid constraints, rigid body initial kinematics, and rigid connectors. The motion of a point $\mathbf{x}$ connected to a rigid body is given by $$\begin{equation} \mathbf{x}=\mathbf{r}\left(t\right)+\mathbf{R}\left(t\right)\cdot\mathbf{Z}=\mathbf{r}\left(t\right)+\mathbf{z}\left(t\right)\label{eq:rbm-position} \end{equation}$$ where $\mathbf{r}\left(t\right)$ is the position of the rigid body center of mass and $\mathbf{R}\left(t\right)$ is the body's (proper orthogonal) rotation tensor, which satisfies $\mathbf{R}\left(t_{0}\right)=\mathbf{I}$ at the initial time $t_{0}$ ; here, $\mathbf{z}\left(t\right)=\mathbf{R}\left(t\right)\cdot\mathbf{Z}$ is the position vector of the point relative to the body's center of mass, and $\mathbf{X}=\mathbf{r}\left(t_{0}\right)+\mathbf{Z}$ is the initial position. In FEBio, the rotation is evaluated as $\mathbf{R}\left(t\right)=\exp\left[\boldsymbol{\xi}\left(t\right)\right]$ , where $\boldsymbol{\xi}\left(t\right)$ is the material rotation of the rigid body from its reference configuration. In practice, $\boldsymbol{\xi}$ is stored as a quaternion to facilitate the multiplication of rotation tensors. The exponential map $\exp\left[\boldsymbol{\xi}\left(t\right)\right]$ is given in full form as $$\begin{equation} \mathbf{R}\left(t\right)=\exp\left[\boldsymbol{\xi}\left(t\right)\right]=\cos\xi\,\mathbf{I}-\sin\xi\,\boldsymbol{\mathcal{E}}\cdot\mathbf{n}+\left(1-\cos\xi\right)\mathbf{n}\otimes\mathbf{n}\label{eq:rbr-rotation-tensor} \end{equation}$$ where $\boldsymbol{\mathcal{E}}$ is the third-order permutation pseudo-tensor with rectangular components $\varepsilon_{ijk}$ (permutation symbol). Here, $\mathbf{n}$ is the unit vector along the axis of rotation and $\xi$ is the angle of rotation about $\mathbf{n}$ , such that $\boldsymbol{\xi}=\xi\mathbf{n}$ . Both $\xi$ and $\mathbf{n}$ may evolve over time. Either none or all the components of $\boldsymbol{\xi}\left(t\right)$ must be prescribed (or fixed) for a rigid body.