The multi-jointed autonomous underwater vehicle (AUV) has higher manoeuvrability and a smaller turning radius in comparison with an AUV with a rigid body. It is able to implement spatially denser sample collection in three-dimensional space in the deep ocean for scientific expeditions. As conventional positioning approaches, such as ultra-short-baseline and long-baseline, cannot be employed to obtain the coordinates of a multi-jointed AUV, because they only work with constant baselines, a novel underwater acoustic positioning system with a single beacon and a varied baseline is proposed. The coordinate transformation based on the angles of rotating joints and the geometry of the multi-jointed AUV is mathematically modelled. Generalised cross-correlation is applied to measure the time delays of sound signals between the beacon and the hydrophones on the AUV. Numerical simulations and experiments in an anechoic water tank were carried out, and the impacts of parameters on the accuracy of the estimated coordinates of the multi-jointed AUV are examined. Accuracies of 0.86% in simulation and 0.2 m (at a range of 30 m) in experiments are demonstrated.