  We investigate the configuration space $\mathcal{S}_{G,b,\ell}$ associated
with the movement of a robotic arm of length $\ell$ on a grid over an
underlying graph $G$, anchored at a vertex $b \in G$. We study an associated
PIP (poset with inconsistent pairs) $\text{IP}_{G,b,\ell}$ consisting of
indexed paths on $G$. This PIP acts as a combinatorial model for the robotic
arm, and we use $\text{IP}_{G,b,\ell}$ to show that the space
$\mathcal{S}_{G,b,\ell}$ is a CAT(0) cubical complex, generalizing work of
Ardila, Bastidas, Ceballos, and Guo. This establishes that geodesics exist
within the configuration space, and yields explicit algorithms for moving the
robotic arm between different configurations in an optimal fashion. We also
give a tight bound on the diameter of the robotic arm transition graph (the
maximal number of moves necessary to change from one configuration to another)
and compute this diameter for a large family of underlying graphs $G$.
