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If the point $\mathbf{x}$ is connected to a rigid body, its motion is given by $$\begin{equation} \mathbf{x}=\mathbf{r}\left(t\right)+\boldsymbol{\Lambda}\left(t\right)\cdot\mathbf{Z}=\mathbf{r}\left(t\right)+\mathbf{z}\left(t\right)\label{eq:rbm-position} \end{equation}$$ where $\mathbf{r}\left(t\right)$ is the position of the rigid body center of mass and $\boldsymbol{\Lambda}\left(t\right)$ is the body's rotation tensor, which satisfies $\boldsymbol{\Lambda}\left(t_{0}\right)=\mathbf{I}$ at the initial time $t_{0}$ ; here, $\mathbf{z}\left(t\right)=\boldsymbol{\Lambda}\left(t\right)\cdot\mathbf{Z}$ is the distance of the point from the body's center of mass, and $\mathbf{X}=\mathbf{r}\left(t_{0}\right)+\mathbf{Z}$ is the initial position. The velocity of that point is $$\begin{equation} \begin{aligned}\dot{\mathbf{x}} & =\dot{\mathbf{r}}\left(t\right)+\dot{\boldsymbol{\Lambda}}\left(t\right)\cdot\mathbf{Z}\end{aligned} \,,\label{eq:rbm-velocity} \end{equation}$$ where $$\begin{equation} \dot{\boldsymbol{\Lambda}}\left(t\right)=\hat{\boldsymbol{\omega}}\left(t\right)\cdot\boldsymbol{\Lambda}\left(t\right)=\boldsymbol{\Lambda}\left(t\right)\cdot\hat{\mathbf{W}}\left(t\right)\,.\label{eq:rbm-lambda-dot} \end{equation}$$ Here, $\hat{\boldsymbol{\omega}}$ is an antisymmetric tensor with axial vector $\boldsymbol{\omega}$ which represents the spatial angular velocity vector; similarly, $\hat{\mathbf{W}}$ is an antisymmetric tensor with axial vector $\mathbf{W}$ (the material angular velocity), such that $\boldsymbol{\omega}=\boldsymbol{\Lambda}\cdot\mathbf{W}$ and $$\begin{equation} \hat{\boldsymbol{\omega}}=\boldsymbol{\Lambda}\cdot\hat{\mathbf{W}}\cdot\boldsymbol{\Lambda}^{T}\,.\label{eq:rbm-material-spatial-angular-velocity} \end{equation}$$ We may now rewrite $$\begin{equation} \begin{aligned}\dot{\mathbf{x}} & =\dot{\mathbf{r}}\left(t\right)+\hat{\boldsymbol{\omega}}\left(t\right)\cdot\mathbf{z}\left(t\right)\\ & =\dot{\mathbf{r}}\left(t\right)+\boldsymbol{\Lambda}\left(t\right)\cdot\hat{\mathbf{W}}\left(t\right)\cdot\mathbf{Z} \end{aligned} \,,\label{eq:rbm-rigid-point-velocity} \end{equation}$$ so that the acceleration of the point is $$\begin{equation} \begin{aligned}\ddot{\mathbf{x}} & =\ddot{\mathbf{r}}\left(t\right)+\left(\hat{\boldsymbol{\alpha}}\left(t\right)+\hat{\boldsymbol{\omega}}^{2}\left(t\right)\right)\cdot\mathbf{z}\\ & =\ddot{\mathbf{r}}\left(t\right)+\boldsymbol{\Lambda}\left(t\right)\cdot\left(\hat{\mathbf{A}}\left(t\right)+\hat{\mathbf{W}}^{2}\left(t\right)\right)\cdot\mathbf{Z} \end{aligned} \,,\label{eq:rbm-rigid-point-acceleration} \end{equation}$$ where $\boldsymbol{\alpha}=\dot{\boldsymbol{\omega}}=\boldsymbol{\Lambda}\cdot\mathbf{A}$ is the spatial angular acceleration vector, $\mathbf{A}=\dot{\mathbf{W}}$ is the material angular acceleration vector. As shown below, the time discretization is performed in the material frame.