The　trajectory　tracking　and　flexible　vibration　suppression　of　a　two-flexible-link　free-floating　space　robot　system　were　discussed　based　on　finite　time　with　the　joint　torque　output　dead-zone.　Using　singular　perturbation　theory,　the　dynamics　equations　of　the　system　were　decomposed　into　a　slow　subsystem　for　rigid　motions　and　a　fast　subsystem　for　flexible　vibrations.　For　the　slow　subsystem　with　unknown　model　parameters　and　dead-zone　parameters,　a　dead-zone　pre-compensator　and　a　finite-time　controller　were　designed　based　on　nominal　model.　An　integral　sliding　mode　surface　with　finite　time　convergence　property　was　introduced,　which　had　faster　convergence　speed,　better　robustness　and　anti-interference　performance　than　the　traditional　asymptotic　convergence　control　method.　For　the　fast　subsystem,　the　linear　quadratic　optimal　control　method　was　adopted　to　suppress　the　vibrations　actively　to　ensure　good　stability　of　the　system.　Combined　with　the　finite-time　stability　lemma,　the　Lyapunov　theory　was　used　to　prove　that　the　proposed　control　algorithm　may　make　the　tracking　errors　converge　to　the　origin　within　a　finite　time.　The　simulation　examples　confirm　the　effectiveness　of　the　proposed　method.　©　2019,　China　Mechanical　Engineering　Magazine　Office.　All　right　reserved.