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The reaction moment $\mathbf{m}^{\left(1\right)}$ generated by a torsional spring acting on rigid body $\left(1\right)$ is given by $$\begin{equation} \boxed{\mathbf{m}^{\left(1\right)}=\boldsymbol{\mu}+\varepsilon_{r}\boldsymbol{\gamma}}\label{eq:jm-joint-moment} \end{equation}$$ where $$\begin{equation} \boxed{\boldsymbol{\gamma}=\frac{1}{2}\sum_{j=1}^{k}\mathbf{e}_{j}^{\left(1\right)}\times\mathbf{e}_{j}^{\left(2\right)}}\label{eq:jm-angular-gap} \end{equation}$$ is the angular gap function between the bases of rigid bodies $\left(1\right)$ and $\left(2\right)$ , $\boldsymbol{\mu}$ is the (optional) Lagrange multiplier used when invoking the augmented Lagrangian method, and $\varepsilon_{r}$ is a penalty parameter representing the torsional spring stiffness. We consider three cases for the choice of $\boldsymbol{\gamma}$ : $$\begin{equation} \boldsymbol{\gamma}=\begin{cases} k=3 & \text{constrain relative rotation along all directions}\\ k=1 & \text{maintain free relative rotation along }\mathbf{e}_{1}^{\left(i\right)}\\ k=0 & \text{maintain free relative rotation along all directions} \end{cases}\label{eq:jm-cases} \end{equation}$$ For example, $k=3$ is used to model the reaction moment in a prismatic joint, whereas $k=1$ is used with revolute, cylindrical and planar joints.