This　paper　focuses　on　the　trajectory　tracking　control　of　an　underactuated　AUV　in　the　vertical　plane.　First,　an　error　analysis　is　conducted　for　the　underactuated　AUV　in　the　vertical　plane,　and　a　Disturbance　Observer　(DO)　is　designed　to　estimate　external　disturbances,　while　a　Radial　Basis　Function　Neural　Network　(RBFNN)　is　employed　to　approximate　nonlinear　terms.　Subsequently,　controllers　are　designed　based　on　Nonsingular　Terminal　Sliding　Mode　Control　(NTSMC)　and　Nonsingular　Fast　Terminal　Sliding　Mode　Control　(NFTSMC),　respectively.　Finally,　the　stability　of　the　controllers　is　verified　using　the　Lyapunov　function.　Simulation　results　demonstrate　that　controllers　effectively　improve　trajectory　tracking　accuracy　and　enhances　the　robustness　of　the　system.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2026.