Structural Controllability of Drift-free Bilinear Systems on $\mathbb{SE(n)}$

We obtain graph theoretic necessary and sufficient conditions for the structural controllability of drift free bilinear systems on the special Euclidean group leveraging results from the existing literature. The graph theoretic conditions allow us to check the structural controllability in polynomial time. We use these conditions to find the sparsest structures for structural controllability. These conditions can also be used to check the structural accessibility of bilinear systems with drift. Equivalent conditions using the permutation group are also obtained. We consider the problem of link failures within a given structure and obtain equivalent conditions for structural controllability which are polynomial time checkable. We show that the problem of finding sparsest structures under $k$ link failures is NP-hard for $k>0$. We also discuss the case of structural controllability under probabilistic link failures.