The　augmented　adaptive　kinematic　control　and　vibration　optimal　control　based　on　velocity　observers　for　a　free-floating　two-link　flexible　space　manipulator　under　external　disturbances　were　studied　when　positions　and　attitudes　of　its　base　were　uncontrollable.　Firstly,　after　selecting　an　appropriate　coordinate　system,　the　dynamic　equations　of　the　free-floating　two-link　flexible　space　manipulator　were　established　with　the　momentum　conservation　principle　and　Lagrange　equations.　Secondly,　using　the　singular　perturbation　approach,　the　manipulator　system　was　divided　into　two　sub-systems　including　a　slowly　varying　subsystem　of　base　attitudes　and　joints＇　trajectory　tracking　and　a　fast　varying　subsystem　to　describe　flexible-links＇　vibration.　Then　a　composite　controller　consisting　of　a　slowly　varying　control　component　and　a　fast　varying　control　component　was　proposed.　The　slowly　varying　subsystem＇s　observed　velocity　vector　was　obtained　with　an　adaptive　sliding　model　velocity　observer.　Based　on　this　observed　velocity　vector,　an　augmented　adaptive　slowly　varying　control　law　was　designed　to　realize　joints＇　trajectory　tracking.　The　observed　velocity　vector　of　the　fast　varying　subsystem　was　obtained　with　a　linear　velocity　observer.　Based　on　this　observed　velocity　vector,　the　fast　varying　control　law　was　designed　to　realize　the　vibration　optimal　control　of　flexible　links　using　the　optimal　linear　quadratic　regulator　(LQR)　method.　Finally,　the　numerical　simulation　of　the　system　verified　the　effectiveness　of　the　proposed　method.　The　results　showed　that　this　control　scheme　does　not　need　direct　measurement　of　position,　moving　velocity,　and　moving　acceleration　of　base,　angular　velocities,　and　angular　accelerations　of　joints　as　well　as　derivatives　of　flexible　vibration　modal　coordinates.　©　2017,　Editorial　Office　of　Journal　of　Vibration　and　Shock.　All　right　reserved.