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<p>This paper investigates the kinematics of a spherical 3-RRR mechanism with synchronously driven cranks, referred to as the spherical F-mechanism. Because the three input cranks are coupled by a common driving law, the mechanism operates as a one-degree-of-freedom spherical parallel system and exhibits constrained but nontrivial motion on the unit sphere. In contrast to the planar case, the spherical setting leads to coupled nonlinear constraints on <i>SO</i>(3), which require a different analytical treatment. To address this, the constraint equations are formulated in terms of unit quaternions, providing a globally valid orientation representation for position analysis and for the formulation of the forward position problem while avoiding the parametrization singularities associated with Euler-angle descriptions. On this basis, a velocity analysis is developed using the geometry of the coupler great circles and their associated normals. This leads to a geometric singularity criterion: the mechanism is singular if and only if the three great circles supporting the coupler arcs intersect at a common pair of antipodal points on the unit sphere. The study also establishes the existence of permanent stillstand modes and derives the corresponding geometric conditions under which branch switching into a nontrivial spatial motion may occur. The theoretical results are illustrated by representative numerical examples and workspace trajectories. Taken together, the results indicate that the spherical F-mechanism preserves the central paradoxical-motion features of its planar counterpart while revealing kinematic effects that arise naturally from spherical constraint geometry.</p>

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Quaternion-based kinematic analysis of a spherical F-mechanism with geometric singularity characterization

  • 期刊论文

  • 发表日期：

    2026年8月4日

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  • Engin Can

摘要

This paper investigates the kinematics of a spherical 3-RRR mechanism with synchronously driven cranks, referred to as the spherical F-mechanism. Because the three input cranks are coupled by a common driving law, the mechanism operates as a one-degree-of-freedom spherical parallel system and exhibits constrained but nontrivial motion on the unit sphere. In contrast to the planar case, the spherical setting leads to coupled nonlinear constraints on SO(3), which require a different analytical treatment. To address this, the constraint equations are formulated in terms of unit quaternions, providing a globally valid orientation representation for position analysis and for the formulation of the forward position problem while avoiding the parametrization singularities associated with Euler-angle descriptions. On this basis, a velocity analysis is developed using the geometry of the coupler great circles and their associated normals. This leads to a geometric singularity criterion: the mechanism is singular if and only if the three great circles supporting the coupler arcs intersect at a common pair of antipodal points on the unit sphere. The study also establishes the existence of permanent stillstand modes and derives the corresponding geometric conditions under which branch switching into a nontrivial spatial motion may occur. The theoretical results are illustrated by representative numerical examples and workspace trajectories. Taken together, the results indicate that the spherical F-mechanism preserves the central paradoxical-motion features of its planar counterpart while revealing kinematic effects that arise naturally from spherical constraint geometry.