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

Compliant mechanisms are important in robotics because they can improve adaptability, safety, and energy efficiency while reducing hardware complexity. This paper presents SPiralRoll, a novel torsion-spring-based underactuated compliant mechanism for rolling robots and compliant robotic actuation. The mechanism uses arc-distributed elastic members and two motor inputs to realize three physically observable output motions: rotational motion, radial expansion/contraction, and axial spin induced by nonlinear compliant deformation. Two configurations, namely full-arc and single-arc designs, are developed and experimentally evaluated. Beyond benchtop validation, the mechanism is integrated into a spherical rolling robot, where proof-of-concept experiments demonstrate forward rolling and turning. The results show that the full-arc design provides better structural support and smoother deformation, whereas the single-arc design yields larger deformation and stronger inertial excitation, making it more suitable for pendulum-driven rolling locomotion. Overall, SPiralRoll provides a low-cost, compact, and fully 3D-printable solution for underactuated compliant rolling robots and adaptive robotic joints.

Rolling motion planning is challenging because rolling contact imposes nonholonomic constraints and the configuration evolves on a curved manifold. The problem becomes substantially harder in multi-contact settings, where multiple bodies roll without slip and the contact states are coupled. This paper presents a new framework for multi-contact path planning in spherical rolling robotics under no-slip constraints. We first derive a compact kinematic model for multi-sphere rolling using Montana's contact-coordinate formulation, where each contact is represented by a stacked five-state vector. Building on this model, we construct a Voronoi-based roadmap directly on the spherical contact manifold, incorporating spherical-cap obstacles and mutual-exclusion regions via on-manifold collision checking, and refine discrete graph paths using manifold-consistent log-exp smoothing. The resulting smoothed surface paths are then lifted to admissible multi-contact rolling motions through the derived Montana kinematics and validated via forward simulation. We further evaluate feasibility and path quality versus trajectory smoothness, Voronoi seed density, and computation time. The proposed framework provides a foundation for extending the method to non-spherical geometries, time-varying obstacle environments, and experimental validation on physical rolling robotic platforms.

An asymmetric two-link robot supported atop a flat platform by wheels that roll and pivot freely, but do not slip laterally, will develop forward momentum if the joint between the links is actuated internally. In particular, oscillations in the joint angle will generate undulatory locomotion suggesting fishlike swimming. If two such robots surmount a common platform that's free to translate with its own inertial dynamics, then the individual robots' dynamics will be coupled so that the locomotion of either robot is affected by that of the other. We develop a mathematical model for this system and present simulations demonstrating its behavior. We then consider a single robot with an unactuated joint rolling atop a platform that moves under control, and show that actuation of the platform is sufficient to dictate the robot's behavior. In particular, with the acceleration of the platform as an input, the robot's heading can be made to track a chosen function of time. This is sufficient to guarantee that the robot can be induced to orbit a fixed point on the platform or to locomote persistently in a desired direction.