Natural Locomotion: Principle and Method

2026-05-27

AI Summary

This work proposes the principle of “natural motion” to overcome robots’ reliance on predefined trajectories, modeling locomotion as an energy exchange process between internal oscillators and environmental constraints, with the requirement that net power exchange over a cycle vanishes. The key contributions include the first formalization of the energy exchange mechanism underlying natural motion, the introduction of the Natural Locomotion Manifold (NLM) to characterize feasible motion patterns, and a design framework grounded in energy conservation and internal recurrence that reframes motion generation as a passive structural design problem. The approach integrates action-angle variables, nonlinear modal analysis, and nonholonomic dynamics, implemented via closed- and open-channel construction methods. Efficient, self-sustained periodic motions are demonstrated on the Chaplygin sleigh–pendulum-driven cart and its three-body extension, along with theoretical criteria for the existence of NLM families.

Abstract

Robotic locomotion can become efficient when mechanisms exploit passive dynamics, compliance, and resonance rather than track prescribed trajectories. This paper formulates natural locomotion as an exchange principle for systems whose motion is mediated by environmental constraints or interactions. A motion is natural when an internal oscillator returns periodically, the body pose drifts, and the mean Propulsion--Oscillator Exchange power (POE power) vanishes over one cycle. The selected family is a Natural Locomotion Manifold (NLM). We develop the conservative realization of this principle for continuous ideal environmental constraints: the constraints do no external work, total mechanical energy is conserved, and zero mean POE power is an internal exchange with the environment-mediated propulsive channel, not external energy input. The method is a closed/open construction. The propulsive channel is first closed to reveal an effective internal oscillator, organized by scalar action-angle structure in one effective degree of freedom or by nonlinear modal sectors in several degrees of freedom. The channel is then reopened, pose is reconstructed, and accepted cycles must preserve internal recurrence and zero mean POE power. We demonstrate the principle on two ideal nonholonomic no-slip systems: a Chaplygin-sleigh / pendulum-driven car and a three-body extension. In the scalar case, POE closure is equivalent to the missing internal return condition, giving a theorem-backed computation of the NLM family. In the multi-degree case, POE closure remains necessary but must be completed by modal identity, internal return, dynamics consistency, same fixed passive architecture, and nonzero displacement. Natural locomotion becomes a design question: which passive architectures support no, one, or several certified NLM families?

Problem

• natural locomotion
• passive dynamics
• Propulsion-Oscillator Exchange power
• nonholonomic systems
• environmental constraints

Innovation

• Natural Locomotion
• Propulsion–Oscillator Exchange (POE) power
• Natural Locomotion Manifold (NLM)
• Passive dynamics
• Nonholonomic systems