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Autonomous Underwater Vehicles find extensive applications in defense organizations for underwater mine detection and region surveillance. These are also useful for oil and gas industries in detection of leakage in the pipelines and also in many other marine industries. Underwater Robots can be categorized into two types namely (i) Remotely Operated Vehicle (ROV) and (ii) Autonomous Underwater Vehicle (AUV). A ROV is a remotely operated vehicle usually connected with the mother ship or base station through a tethered wire whereas AUV is an Autonomous Underwater Vehicle which traverses autonomously without any external interference. As opposed to ROV, control of an AUV is difficult because it is an underactuated system (whose actuator inputs are less than the number of degrees of freedom to be controlled), also the dynamics of AUV is influenced by external disturbances such as ocean current and hydrodynamic effects. The motion control problems of an AUV can be of different types such as path following, trajectory tracking, waypoint tracking and also localization. The thesis first develops path following control of a single AUV using the Serret- Frenet(S-F) frame approach and error backstepping technique. Later on the same back- stepping approach has been extended for implementation of formation control for multiple AUVs. Out of various motion control strategies, this thesis mainly focusses on path following control problem of a single AUV. To address this problem of path following, a virtual frame is considered. This virtual moving frame is called the S-F frame. The purpose of using S-F frame is to represent the AUV …