The　dynamic　modeling,　motion　control　and　flexible　vibration　active　suppression　of　space　robot　under　the　influence　of　flexible　base,　flexible　link　and　flexible　joint　are　explored,　and　motion　and　vibration　integrated　fixed-time　sliding　mode　control　of　fully　flexible　system　is　designed.　The　flexibility　of　the　base　and　joints　are　equivalent　to　the　vibration　effect　of　linear　springs　and　torsion　springs.　The　flexible　links　are　regarded　as　Euler-Bernoulli　simply　supported　beams,　which　are　analyzed　by　the　hypothetical　mode　method,　and　the　dynamic　model　of　the　fully　flexible　space　robot　is　established　by　using　the　Lagrange　equation.　Then,　the　singular　perturbation　theory　is　used　to　decompose　the　model　into　slow　subsystem　including　rigid　motion　and　the　link　flexible　vibrations,　and　fast　subsystems　including　the　base　and　the　joint　flexible　vibrations.　A　fixed　time　sliding　mode　control　based　on　hybrid　trajectory　is　designed　for　the　slow　subsystem　to　ensure　that　the　base　and　joints　track　the　desired　trajectory　in　a　limited　time　while　achieving　vibration　suppression　on　the　flexible　links.　For　the　fast　subsystem,　linear　quadratic　optimal　control　is　used　to　suppress　the　flexible　vibration　of　the　base　and　joints.　The　simulation　results　show　that　the　controller　proposed　in　the　paper　can　make　the　system　state　converge　within　a　fixed　time,　is　robust　to　model　uncertainty　and　external　interference,　and　can　effectively　suppress　the　flexible　vibration　of　the　base,　links,　and　joints.