The　trajectory　tracking　and　vibration　suppression　of　elastic　base　and　two　flexible　arms　for　a　free-floating　space　robot　system　with　flexible　arms　and　elastic　base　is　discussed.　Using　singular　perturbation　theory　of　two　kinds　of　time　scale,　a　slow-subsystem　describing　the　rigid　motion　and　a　fast-subsystem　corresponding　to　vibration　of　base　elastic　and　two　flexible　arms　are　obtained.　For　the　slow-subsystem,　neural　network　is　used　to　approximate　the　two　uncertainties　in　the　dynamic　model　of　the　space　robot.　The　integral　term　of　proportional　integral　(PI)　sliding　mode　control　makes　initial　state　of　the　system　falls　on　the　sliding　surface,　which　can　offset　the　unknown　disturbance　at　the　beginning　of　control　and　eliminate　the　chattering　of　controller　output,　so　the　robustness　of　the　control　law　is　enhanced.　The　PI　sliding　mode　neural　network　controller　is　designed　to　improve　the　tracking　performance　of　the　rigid　point-to-point　movement.　For　the　fast-subsystem,　an　optimal　linear　quadratic　regulator(LQR)　controller　is　adopted　to　damps　out　the　vibration　of　the　two　flexible　links　and　base　elastic,　which　guarantees　the　stability　and　tracking　accuracy　of　the　system.　Finally,　computer　simulation　results　show　the　effectiveness　of　the　compound　control　method.　©　2018　Published　under　licence　by　IOP　Publishing　Ltd.