Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments

Abstract

Coordinate singularities and gimbal lock are two phenomena that present themselves in models for the dynamics of mechanical systems. The former phenomenon pertains to the coordinates used to parameterize the configuration manifold of the system, while the latter phenomenon has a distinctive physical manifestation. In the present paper, we use tools from differential geometry to show how gimbal lock is intimately associated with an orthogonality condition on the applied forces and moments which act on the system. This condition is equivalent to a generalized applied force being normal to the configuration manifold of the system. Numerous examples, including the classic bead on a rotating hoop example and a gimbaled rigid body, are used to illuminate the orthogonality condition. These examples help to offer a new explanation for the elimination of gimbal lock by the addition of gimbals and demonstrate how integrable constraints alter the configuration manifold and may consequently eliminate coordinate singularities.