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

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.