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Deploying a multi-hop underwater acoustic sensor network (UASN) in a large area brings about new challenges in reliable data transmissions and survivability of network due to the limited underwater communication range/bandwidth and the limited energy of underwater sensor nodes. In order to address those challenges and achieve the objectives of maximization of data delivery ratio and minimization of energy consumption of underwater sensor nodes, this paper proposes a new underwater routing scheme, namely AURP (AUV-aided underwater routing protocol), which uses not only heterogeneous acoustic communication channels but also controlled mobility of multiple autonomous underwater vehicles (AUVs). In AURP, the total data transmissions are minimized by using AUVs as relay nodes, which collect sensed data from gateway nodes and then forward to the sink. Moreover, controlled mobility of AUVs makes it possible to apply a short-range high data rate underwater channel for transmissions of a large amount of data. To the best to our knowledge, this work is the first attempt to employ multiple AUVs as relay nodes in a multi-hop UASN to improve the network performance in terms of data delivery ratio and energy consumption. Simulations, which are incorporated with a realistic underwater acoustic communication channel model, are carried out to evaluate the performance of the proposed scheme, and the results indicate that a high delivery ratio and low energy consumption can be achieved.