Natural efficient gaits from Nonholonomic Locomotion Nonlinear Normal Mode (NL-NNM): The Pendrivencar case

Abstract

Bio-inspired robots remain far less energy-efficient than animals because conventional controllers impose trajectories that fight passive dynamics, whereas animals exploit resonance through natural nonlinear normal modes (NNM), whose periodic internal motions form a smooth 2D invariant surface; We ask how to define and compute the natural motions of a conservative locomotion system: propulsion arises only from no-slip constraints, and once initiated, a gait persists without actuation—like a frictionless pendulum. We tackle non-holonomic constraints on the Pendrivencar, a vehicle driven by a motorised pendulum with a cubic torsional spring; We introduce the Nonholonomic Locomotion - NNM (NL-NNM): extract a high-speed spectral seed – where chassis oscillations vanish and the pendulum is neutrally stable – refine the periodic orbit, and continue the resulting 2D invariant manifold via pseudo-arclength across three slow centre manifolds (stable for positive speed, neutral at zero, unstable for negative) from non-isolated rectilinear equilibria; We demonstrate the first NL-NNM for a moving non-holonomic robot: internal orbits produce a pendulum–chassis choreography whose energy-dependent frequency shifts and harmonic richness exceed linear predictions. Via geometric phase, each orbit yields undulatory straight-line motion. A dual-loop control simulation confirms autonomous path tracking with only the pendulum; Extending to dissipative regimes via non-linear resonant modes offers a path to high-efficiency locomotion in aquatic, aerial, legged, soft-bodied, and other robots.