Full Nonlinear Nonholonomic Dynamics and Motion Analysis of a 3-DoF Underactuated Spherical Rolling Robot

2026-06-20

AI Summary

This study addresses the absence of a complete nonlinear nonholonomic dynamic model for MonoRollBot, a single-motor-driven three-degree-of-freedom underactuated spherical robot. By applying the Lagrange–d’Alembert principle and incorporating both the holonomic constraints from the screw pair and the Pfaffian nonholonomic constraints arising from pure rolling without slip, this work establishes—for the first time—a full-state nonlinear dynamic model that simultaneously captures the shell’s translation and orientation, screw displacement, nut rotation, and radial mass motion. The formulation retains all generalized coordinates and explicitly reveals the coupling mechanisms among gravity, compliance, and inertia that govern the robot’s mobility. Qualitative agreement between simulations and physical prototype experiments validates the model’s fidelity, thereby providing a mechanically consistent dynamical foundation for subsequent state estimation and hybrid control strategies.

Abstract

This paper presents a full nonlinear constrained dynamic model of MonoRollBot, a novel 3-DoF spherical rolling robot driven by a single motor, a lead-screw transmission, and a spring-coupled internal moving mass, together with motion analysis of its behavior. To the best of our knowledge, this is one of the first full nonlinear nonholonomic models reported for a mono-actuated, super-underactuated spherical rolling robot of this kind. Because rolling without slipping is nonholonomic, the dynamics are derived using the Lagrange--d'Alembert formulation, with the lead-screw relation imposed as a holonomic constraint and the rolling condition imposed in Pfaffian form. The formulation retains the complete generalized coordinates of shell translation, shell attitude, screw travel, nut rotation, and radial mass motion. Simulations and representative motion studies show qualitative agreement with prototype behavior and reveal how gravity, compliance, and inertia jointly shape the locomotion and motion capabilities of this strongly underactuated robot. The resulting model also provides a mechanically consistent basis for future state estimation and hybrid controller design for this nonholonomic mono-actuated rolling robot.

Problem

• nonholonomic dynamics
• underactuated spherical robot
• rolling without slipping
• nonlinear modeling
• motion analysis

Innovation

• nonholonomic dynamics
• underactuated spherical robot
• Lagrange–d'Alembert formulation
• mono-actuated system
• rolling without slipping