A　hybrid-trajectory　based　terminal　sliding　mode　controller　(TSMC)　is　addressed　for　a　free-flying　two-flexible-link　space　manipulator　with　an　elastic　base.　In　this　system,　there　are　unknown　but　bounded　external　disturbances　and　parameters.　First,　the　Lagrange　dynamic　model　of　the　manipulator　was　established　by　the　momentum　conservation.　Second,　a　TSMC　based　on　desired　trajectory　was　proposed,　by　which　the　terminal　trajectories　were　asymptotically　tracked　and　periodic　flexible　vibrations　were　excited.　Then　based　on　virtual　control　force,　hybrid　trajectories　were　generated,　in　which　the　flexible　variables,　the　joint　angular　displacement　errors　and　the　base＇s　attitude　error　were　considered.　Finally,　a　hybrid-trajectory　TSMC　was　presented,　by　which　the　terminal　trajectories　were　asymptotically　tracked　and　the　flexible　vibrations　were　suppressed.　©　Cambridge　University　Press　2019.