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Consider a contractile force between points $\mathbf{x}^{\left(i\right)}$ on rigid bodies $\left(1\right)$ and $\left(2\right)$ . The force is oriented along $\mathbf{n}$ , where $$\begin{equation} \mathbf{n}=\frac{\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}}{\left|\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\right|}\,.\label{eq:rcf-1} \end{equation}$$ Let $L=\left|\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\right|$ and $L_{0}=\left|\mathbf{X}^{\left(2\right)}-\mathbf{X}^{\left(2\right)}\right|$ , so that the contractile force acting on body $\left(1\right)$ is given by $$\begin{equation} \mathbf{f}^{\left(1\right)}=f_{0}\mathbf{n}=\frac{f_{0}}{L}\left(\mathbf{x}^{\left(2\right)}-\mathbf{x}^{\left(1\right)}\right)\,.\label{eq:rcf-2} \end{equation}$$ We do not need to use the augmented Lagrangian method here, thus