The compliance of a fin affects the thrust of underwater vehicles mimicking the undulatory motion of fish. Determining the optimal compliance of a fin to maximize thrust is an important issue in designing robotic fish using a compliant fin. We present a simple method to identify the condition for maximizing the thrust generated by a compliant fin propulsion system. When a fin oscillates in a sinusoidal manner, it also bends in a sinusoidal manner. We focus on a particular kinematic parameter of this motion: the phase difference between the sinusoidal motion of the driving angle and the fin-bending angle. By observing the relationship between the thrust and phase difference, we conclude that while satisfying the zero velocity condition, the maximum thrust is obtained when compliance creates a phase difference of approximately π/2 at a certain undulation frequency. This half-pi phase delay condition is supported by thrust measurements from different compliant fins (four caudal-shaped fins with different aspect ratios) and a beam bending model of the compliant fin. This condition can be used as a guideline to select the proper compliance of a fin when designing a robotic fish.
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