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

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