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When a joint has a translational degree of freedom along $\mathbf{n}^{\left(1\right)}$ , there may be a need to prescribe the relative displacement $d$ along that direction. This can be achieved by supplementing the joint reaction force with a force that closes the gap between the current and desired translation, $$\begin{equation} \mathbf{f}^{\left(1\right)}=\varepsilon_{c}\left[\left(\mathbf{n}^{\left(1\right)}\otimes\mathbf{n}^{\left(1\right)}\right)\cdot\mathbf{g}-d\,\mathbf{n}^{\left(1\right)}\right]=\varepsilon_{c}\left(\mathbf{P}_{3}\cdot\mathbf{g}-d\,\mathbf{n}^{\left(1\right)}\right)\,,\label{eq:pdj-1} \end{equation}$$ where $\mathbf{P}_{3}$ is given in (7.9.4.1-3). Then, $$\begin{equation} D\mathbf{f}^{\left(1\right)}=D\mathbf{P}_{3}\cdot\mathbf{g}+\mathbf{P}_{3}\cdot D\mathbf{g}+d\,\hat{\mathbf{n}}^{\left(1\right)}\cdot\Delta\boldsymbol{\theta}^{\left(1\right)}\label{eq:pdj-2} \end{equation}$$ Using the results of Section 7.9.4.1↑, the stiffnesses are supplemented with $$\begin{equation} \boxed{\begin{aligned}\mathbf{K}_{fr}^{\left(1\right)} & =-\varepsilon_{c}\mathbf{P}_{3}\\ \mathbf{K}_{f\theta}^{\left(1\right)} & =\varepsilon_{c}\left(\mathbf{P}_{3}\cdot\hat{\mathbf{z}}^{\left(1\right)}+\mathbf{Q}+d\,\hat{\mathbf{n}}^{\left(1\right)}\right)\\ \mathbf{K}_{fr}^{\left(2\right)} & =\varepsilon_{c}\mathbf{P}_{3}\\ \mathbf{K}_{f\theta}^{\left(2\right)} & =-\varepsilon_{c}\mathbf{P}_{3}\cdot\hat{\mathbf{z}}^{\left(2\right)} \end{aligned} }\,.\label{eq:pdj-3} \end{equation}$$