Strapdown inertial navigation systems are widely used in surface ships and warships. Although high-precision optical fibre inertial navigation systems are available, they have high cost and limited practicality. Therefore, they cannot replace the traditional platform inertial navigation systems in all ships. Hence, microelectromechanical system (MEMS)-based inertial sensors are widely used for robust navigation. Accurate and timely identification of sensor faults while ensuring stable navigation is a challenging task. This paper proposes a robust fault detection (FD) approach for a low-cost system that loosely integrates a strapdown inertial navigation system and the global navigation satellite system, where the integrated navigation state estimation provides high-accuracy output. A cubature Hi/H∞-optimised FD filter was designed for a nonlinear discrete time-varying system considering sensitivity to faults and robustness to disturbances. Furthermore, a threshold for FD was derived considering a compromise between the false alarm rate and fault diagnosis accuracy. Finally, the proposed method was validated through simulations using multiple noise distribution sensor data generated by a ship-manoeuvring simulator.