A　dynamics　modeling,　motion　controller　design　and　joint-link　double　flexible　vibration　suppression　problem　of　free-floating　flexible-link　and　flexible-joint　space　robot　were　discussed.　The　dynamic　equations　of　the　system　were　established　according　to　system＇s　momentum　conservation,　angular　momentum　conservation　and　Lagrange-assumed　mode　method.　Based　on　singular　perturbation　method,　the　system　was　decomposed　into　three　mutual　independent　subsystems:　a　＂rigid-joint　and　rigid-link＂　slow　subsystem　represented　the　rigid-joint　and　rigid-link　system＇s　rigid　motion,　a　＂flexible-joint　and　rigid-link＂　fast　subsystem　represented　the　system＇s　elastic　vibration　caused　by　flexible-joint　and　a　＂rigid-joint　and　flexible-link＂　fast　subsystem　represented　the　system＇s　elastic　vibration　caused　by　flexible-link.　Then　three　controllers　were　designed　for　the　three　subsystems　respectively.　A　nonlinear　sliding　mode　controller　was　designed　to　realize　the　asymptotic　tracking　of　the　system＇s　motion　trajectory.　A　feedback　controller　was　used　to　suppress　the　flexible　vibration　caused　by　flexible-joint　based　on　the　velocity　difference　between　the　link　and　the　motor.　A　linear　quadric　regulator　was　used　to　suppress　the　flexible　vibration　caused　by　flexible-link.　Therefore,　the　system＇s　total　controller　was　a　composite　of　the　three　controllers.　The　simulation　proves　the　proposed　compound　controller　can　guarantee　the　control　system＇s　accuracy　and　suppress　the　elastic　vibration　caused　by　flexible-joint　and　flexible-link　effectively.