This　paper　investigates　the　robust-adaptive　neural　network　finite-time　control　and　vibration　suppression　for　a　two-link　flexible-base　flexible-joint　space　manipulator　with　uncertain　dynamics.　Based　on　the　dynamic　equations,　a　slow　subsystem　and　a　fast　subsystem　of　flexible　space　manipulator　are　derived　by　the　singular　perturbation　method.　For　the　slow　subsystem,　a　robust-adaptive　neural　network　finite-time　motion　control　scheme　is　proposed　to　achieve　the　trajectory　tracking　of　the　manipulator,　taking　into　account　both　parameter　uncertainties　and　external　disturbances.　This　control　scheme　consists　of　a　finite-time　controller　used　to　guarantee　the　finite-time　convergence　of　system　errors,　an　adaptive　neural　network　controller　utilized　to　approximate　uncertain　dynamics　and　a　robust　controller　added　to　eliminate　the　approximation　error　of　the　neural　network.　For　the　fast　subsystem,　a　linear　quadratic　optimal　controller　is　introduced　to　suppress　the　vibration　caused　by　flexible　joints　and　the　flexible　base,　respectively.　The　effectiveness　and　the　superiority　of　the　presented　hybrid　controller　are　indicated　by　the　simulations　with　a　planar　two-link　flexible-base　flexible-joint　space　manipulator.　©　2018　IEEE.