Autonomous Underwater Vehicles (AUVs) are widely used in commercial, military and scientific missions for various purposes. Fault Detection and Isolation (FDI) technology is urgently required because of the long duration of missions and the unknown environment. It is necessary to detect and isolate the fault to increase the reliability and availability of AUVs during the task-performing process. The Extended Multiple-Model Adaptive Extended Kalman Filter (EMMAEKF) method is widely used in FDI technology, but there are errors in the linearization process of Extended Kalman Filter (EKF). To overcome this limitation, a new nonlinear Extended Multiple-Model Adaptive Cubature Kalman Filter (EMMACKF) method is proposed in this article. The CKF has been used to generate residual signal on the six degrees-of-freedoms (DOFs) model of AUVs. Simulation results have shown that the original states and extended states can be well evaluated under the actuator fault scenario, and the faulty apparatus can be simultaneously detected and effectively isolated using the proposed methods. Compared with the EMMAEKF and Extended Multiple-Model Adaptive Unscented Kalman Filter (EMMAUKF) algorithms, both accuracy and time delay have been improved to some extent.