A　recently　developed　index　reduction　method　is　applied　to　comparatively　complicated　underactuated　mechanical　systems　in　the　context　of　inverse　dynamics.　The　inverse　dynamics　formulation　is　carried　out　by　employing　servo　constraints　in　which　the　outputs　(specified　in　time)　are　expressed　by　state　variables.　The　dynamic　equations　are　governed　by　differential-algebraic　equations　(DAEs)　with　high　index.　If　redundant　coordinates　are　used　to　formulate　servo-constraint　problems,　the　algebraic　equations　in　the　DAEs　contain　both　servo　and　holonomic　constraints.　It　is　highly　challenging　to　solve　the　DAEs　with　high　index.　Hence　index　reduction　approaches　are　required.　Index　reduction　by　minimal　extension　facilitates　the　desired　index　reduction　and　thus　makes　possible　the　stable　numerical　integration　of　the　index-reduced　DAEs.　In　this　paper　we　apply　the　advocated　method　to　a　very　general　and　versatile　formulation　of　comparatively　complicated　crane　systems.　In　contrast　to　other　schemes　previously　developed　for　underactuated　systems　subject　to　servo　constraints,　the　present　approach　makes　feasible　numerically　solving　the　challenging　inverse　dynamics　problems　presented　herein.