Actuator　failure　and　joint　flexibility　will　dramatically　impact　space　robot　system　control.　In　this　paper,　free-floating　flexible-joint　space-manipulator　dynamic-modeling　is　studied　and　a　state-observer-based　robust　backstepping　fault-tolerant　control　is　proposed　for　the　system　joint　actuator　failure.　Based　on　the　flexible-joint　simplified　model,　the　system’s　rigid-flexible　coupled-dynamic　equations　are　established　according　to　momentum　conservation,　angular　momentum　conservation,　and　the　Lagrange　equation.　Then　the　system　is　decoupled　based　on　the　singular　perturbation　method.　For　the　slow　subsystem,　a　robust　backstepping　fault-tolerant　controller　base　on　a　state　observer　is　designed　to　eliminate　the　angle　error,　compensate　for　the　uncertain　parameter　and　the　external　disturbance,　and　achieve　the　joint-trajectory　asymptotic-tracking.　The　use　of　a　speed　filter　makes　it　inappropriate　to　measure　and　provide　feedback　about　the　system’s　velocity　signals,　so　the　controller　is　simpler　and　more　precise.　For　the　fast　subsystem,　a　velocity　differential-feedback　control　is　adopted　to　suppress　the　system　vibration　caused　by　the　flexible　joint,　to　ensure　the　stability　of　the　system.　Finally,　the　feasibility　and　effectiveness　of　the　model　and　control　method　are　proved　by　some　simulations.　The　simulation　results　indicate　that　the　proposed　fault-tolerant　control　method　can　make　the　free-floating　flexible-joint　space　manipulator　system　track　the　desired　trajectory　accurately　and　steadily,　regardless　of　whether　the　actuator　fails　or　not.　©　2023　by　the　authors.