Energy-Efficient Nonholonomic Fish Robot: Nonlinear Forced Oscillations

This paper investigates a forced-oscillation strategy for efficient propulsion in a minimalistic fish-inspired robot, modeled as a two-segment Chaplygin sleigh with a linear torsion spring. We apply a pure sinusoidal torque in open loop under linear viscous damping and use pseudo-arclength continuation to perform a Nonlinear Frequency Response Analysis (NFRA), thereby uncovering multiple coexisting limit cycles for the same forcing parameters enabling forward gaits. Although linear normal-mode approximations can capture small-amplitude oscillations at very low speeds, they fail to account for energy-dependent natural frequency shifts and the increasingly harmonic-rich head-tail “choreographies” observed at higher velocities. A key result is the identification of nonlinear resonance modes, in which the ratio of the robot's maximum forward velocity to the excitation amplitude is maximized. In our numerical study—focused on velocities below 13 mis—these modes exhibit a resonant period shift of up to 3.8 % of the characteristic time (i.e., square root of inertia over stiffness) relative to the linear approach, and crucially reveal how an evolving head-tail coordination (rather than a fixed, purely sinusoidal pattern) enables periodic exchanges of kinetic and elastic potential energy while leaning on the nonholonomic constraint environment (absence of lateral slip) for net locomotion. At resonance, the actuator essentially offsets viscous losses by exploiting these passive dynamics. Notably, a quadrature-phase input-output forcing achieves less than 0.01 % discrepancy in peak velocity compared to the nonlinear resonance—suggesting that future closed-loop quadrature-based controllers could build upon this principle. These findings underscore the promise of elastic compliance and nonholonomic effects in bio-inspired robots, paving the way for energy-efficient marine vehicles through nonlinear resonance exploitation.