To　suppress　the　vibration　generated　by　the　elastic　base　and　flexible　arms　of　a　free-floating　space　robot　and　avoid　being　influenced　by　the　joint-torque-output　dead　zone,　in　this　paper,　we　discuss　the　trajectory　tracking　and　double　vibration　suppression　of　the　floating-space　robot　system　with　a　joint-torque-output　dead　zone.　By　considering　the　elastic　connection　between　the　elastic　base　and　the　link　as　a　linear　spring　and　using　the　assumed　mode　method,　we　model　the　system　dynamics.　Next,　using　the　singular　perturbation　theory,　we　obtain　a　slow　subsystem　that　describes　the　rigid　motion　and　a　fast　subsystem　that　corresponds　to　the　vibration　of　an　elastic　base　and　two　flexible　links.　For　the　slow　subsystem,　we　design　a　dynamic　surface　control　method　based　on　an　adaptive　Gaussian　fuzzy　logic　function　for　solving　the　dead-zone　problem　of　the　system.　To　reduce　the　amount　of　calculation　required,　we　introduce　a　dynamic　surface　control　scheme　that　avoids　the　calculation　expansion　associated　with　the　backstepping　method.　We　apply　a　fuzzy　logic　function　to　approximate　the　uncertainty　term　of　the　dynamic　equation,　including　dead-zone　errors　and　external　disturbances.　For　the　fast　subsystem,　we　adopt　an　optimal　linear　quadratic　controller　to　dampen　any　vibration　of　the　two　flexible　links　and　elastic　base.　Using　Matlab　to　perform　the　simulation,　the　results　show　that　the　designed　control　scheme　makes　the　base　attitude　and　relative　angles　of　the　two　arms　converge　to　the　desired　angles　and　the　base　displacement　and　the　first-　and　second-order　vibration　modes　of　the　arms　converge　to　zero.　©　2019,　Editorial　Department　of　Journal　of　HEU.　All　right　reserved.