A Two-Stage Neural Network Framework for Swimming Direction Control of an Underwater Walking Robot: Kinematic Validation via Simulation and In-Air Experiments

Each year, hundreds of thousands of tons of derelict fishing gear become entangled in ship propellers, and resolving such incidents currently requires divers to manually cut the entangled material underwater, exposing them to the risk of secondary accidents. To address this problem, we developed a radial quadruped underwater robot that walks along the hull of a ship using electromagnetic feet and swims freely using propellers mounted on the distal link of each leg. However, because each leg has only four degrees of freedom (DOFs), the platform is an underactuated system that cannot independently determine the six-DOF position and orientation required to realize an arbitrary thrust direction. Moreover, the underlying inverse problem is non-unique because a given propulsion direction can be achieved by infinite leg-posture combinations. Extending the vector-thrust allocation scheme used in remotely operated vehicles, this study proposes a two-stage artificial neural network framework that resolves these two issues separately. The first-stage network, a mixture density network with K=7 Gaussian mixture components selected by sweeping K from 5 to 12, maps a target swimming direction to a multimodal distribution of foot-tip positions, thereby capturing the multiple valid leg postures without collapsing them into an infeasible average. The second-stage network performs a near one-to-one regression from these foot-tip positions to the corresponding joint angles, which then serve as reference commands tracked by an independent joint-space proportional-derivative controller. The framework was validated kinematically for nine target directions clustered near the vertical (near-upward swimming) axis in a Gazebo simulation and on an in-air test platform. The mean direction tracking error was 4.25° in simulation and 1.53° in the in-air experiments, confirming that the proposed framework reliably realizes the commanded swimming direction under kinematic validation.