The　main　issue　of　this　paper　is　about　the　space　robot　capture　operation.　The　stabilization　control　and　vibration　suppression　control　of　the　rigid-flexible　coupling　space　manipulator　during　capture　debris　is　presented　in　this　paper.　Firstly,　The　dynamical　model　of　the　space　robot　system　is　derived　with　Lagrange　formula,　the　last　link　of　the　robot　is　considered　as　a　flexible　link,　one　end　of　this　link　is　under　the　constraint　of　a　rotating　hinge　which　connect　with　the　previous　rigid　link,　the　other　end　of　this　link　is　combined　with　the　debris　after　capture　operation,　the　inertia　of　the　debris　is　much　bigger　than　this　link,　so　the　flexible　link　is　assumed　to　be　a　simply-supported　beam　base　on　the　assumed　mode　method　of　the　Euler-Bernoulli　beam　theory;　and　the　other　components　of　the　space　robot　is　rigid.　The　debris　is　considered　as　a　rigid　body,　the　dynamic　model　of　the　debris　is　derived　with　the　Newton-Euler　method.　Second,　base　on　the　dynamic　models　of　the　space　robot　and　debris,　the　impact　effect　is　calculated　from　the　momentum　principle　during　the　capture　operation.　After　the　successfully　capture,　the　initial　momentum　of　the　debris　has　a　new　distribution　in　the　space　robot　and　debris　combined　system,　this　will　cause　the　motion　state　of　the　space　robot　and　debris　combined　system　change,　the　attitude　of　the　space　robot　base　will　change,　the　components　of　the　combined　system　will　interfere,　the　flexible　link　will　vibrate,　all　these　will　cause　great　harm　for　the　combined　system　after　capture.　Third,　this　paper　aim　at　the　above　problems　after　the　capture　operation,　the　dynamic　model　of　the　combined　is　derived　which　is　base　on　the　above　dynamic　models　of　the　space　robot　and　debris　before　the　capture.　At　last,　the　robust　control　base　on　the　dynamic　model　is　employed　to　control　the　stabilization　control　of　the　combined　system　after　capture　operation,　and　the　Linear　Quadrics　optimal　control　is　employed　suppress　the　vibration　of　the　flexible　link,　the　above　control　methods　are　employed　at　the　same　time.　The　computer　simulation　is　presented,　the　result　verify　the　feasibility　of　the　above　control　method.　Copyright　©　(2012)　by　the　International　Astronautical　Federation.