This　paper　addresses　the　problem　of　computing　a　time-optimal　trajectory　between　two　configurations　of　a　carlike　robot.　Since　1990,　two　remarkable　results　of　this　problem　have　been　achieved:　the　sufficient　family　containing　all　types　of　optimal　trajectories　between　any　two　configurations,　and　then　the　optimal　synthesis　determining　a　type　of　optimal　trajectories　for　two　specified　configurations.　Our　contribution　is　to　propose　an　algorithm　which　can　determine　a　precise　trajectory　to　link　two　specified　configurations.　Moreover,　this　algorithm　has　three　features:　first　it　can　achieve　an　optimal　trajectory　ending　at　the　exact　final　configuration;　second　it　is　complete,　i.e.　it　allows　to　determine　an　optimal　trajectory　to　link　any　two　configurations;　and　third　it　has　very　low　computational　cost.　The　rules　for　designing　this　algorithm　result　from　a　new　geometric　reasoning　in　the　switching　space,　which　is　another　contribution　of　this　paper.　©2007　IEEE.