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A new actuation principle which permits omnidirectional steering for a swimming robot using a magnetic resonance imaging scanner is presented. The robot fish is made of a ferromagnetic head and a flexible tail. It is actuated by transverse oscillating magnetic gradients. The swimming performances of the robot fish are studied for varying tail length as well as varying actuation frequency and amplitude. Through a dimensional analysis, the important parameters influencing the swimming gait are identified and the mechanism of actuation is better understood. Considering the scaling of forces, this dimensional analysis leads us to believe that in the future the height and width of the fish robot could be miniaturised to sub-millimetre scale.