Toroidal　drives　combine　a　planetary　gear　drive　with　a　worm　gear　drive.　Furthermore,　in　toroidal　drives,　the　planetary　gear　teeth　are　movable.　The　motion　of　spherical　movable　teeth　for　planetary　gears　affects　the　friction,　wear　and　transmission　efficiency　of　the　meshing　surface,　but　this　effect　has　not　been　thoroughly　studied　to　date.　This　study　involved　a　kinematic　analysis　of　planetary　gear　teeth.　Frenet-fram　was　first　introduced　into　the　toroidal　drive　to　describe　the　motion　of　spherical　movable　teeth.　A　contact　analysis　of　the　spherical　movable　teeth　was　carried　out,　and　a　mathematical　model　of　rolling–sliding　motion　between　the　spherical　movable　teeth　and　the　central　worm　spiral　groove　was　established　via　the　conversion　mechanism　method.　The　instantaneous　velocity　formula　and　relative　sliding　velocity　formula　at　the　meshing　points　of　movable　teeth　were　derived.　The　influence　of　the　system　parameters　on　the　sliding　velocity　was　analysed,　and　several　useful　conclusions　were　drawn.　These　results　can　provide　a　theoretical　foundation　for　subsequent　research　on　the　friction　loss　and　transmission　efficiency　of　toroidal　drives.　©　The　Author(s)　2025.