Parametric roll oscillations of a hydrodynamic Chaplygin sleigh

TL;DR

This paper analyzes how parametric excitation affects the roll stability of a fish-like underwater robot modeled as a hydrodynamic Chaplygin sleigh, revealing trade-offs between speed, efficiency, and stability.

Contribution

It introduces a simplified nonholonomic model to study the impact of periodic yaw motion on roll stability using Floquet theory and Mathieu equations.

Findings

Fast swimming motions often lead to roll instability.

Added mass effects influence the linear dynamics and stability.

The model highlights fundamental stability trade-offs in fish-like robots.

Abstract

Biomimetic underwater robots use lateral periodic oscillatory motion to propel forward, which is seen in most fishes known as body caudal fin (BCF) propulsion. The lateral oscillatory motion makes slender-bodied fish-like robots roll unstable. Unlike the case of human-engineered aquatic robots, many species of fish can stabilize their roll motion to perturbations arising from the periodic motions of propulsors. To first understand the origin of the roll instability, the objective of this paper is to analyze the parameters affecting the roll-angle stability of an autonomous fish-like underwater swimmer. Eschewing complex models of fluid-structure interaction, we instead consider the roll motion of a nonholonomic system inspired by the Chaplygin sleigh, whose center of mass is above the ground. In past work, the dynamics of a fish-like periodic swimmer have been shown to be similar to that of a Chaplygin sleigh. The Chaplygin sleigh is propelled by periodic torque in the yaw direction. The roll dynamics of the Chaplygin sleigh are linearized and around a nominal limit cycle solution of the planar hydrodynamic Chaplygin sleigh in the reduced velocity space. It is shown that the roll dynamics are then described as a nonhomogeneous Mathieu equation where the periodic yaw motion provides the parametric excitation. We study the added mass effects on the sleigh's linear dynamics and use the Floquet theory to investigate the roll stability due to parametric excitation. We show that fast motions of the model for swimming are frequently associated with roll instability. The paper thus sheds light on the fundamental mechanics that present trade-offs between speed, efficiency, and stability of motion of fish-like robots.