# A Geometric Perspective on Gimbal Lock in the Apollo program

The phenomenon of gimbal lock rose in prominence during the time  of NASA's Apollo program. Inside the inertial measurement unit of the Apollo spacecraft was a stable (inertial) platform about which the spacecraft rotated. The hinged platform was suspended in two gimbals to give it three degrees of freedom. Its role was to give an accurate measure of the spacecraft's orientation. At a particular orientation of the spacecraft, the gimbals became coplanar and control torques became ineffective at stabilizing the platform. Moreover, if the spacecraft then rotated about an axis normal to the common plane of the gimbals, the platform became ``locked" and engaged in the same rotation as the spacecraft, thereby rendering it useless as an inertial reference. To correct for the locking of the  gimbals, the spacecraft would have to pitch away from the problematic orientation and the platform would be reoriented relative to the stars. To avoid gimbal lock entirely, three gimbals could be used with a control scheme to keep three suspension axes 90 degrees apart. For the Apollo program, the engineers, who were fully cognizant of gimbal lock, opted to use less gimbals to save on weight costs, and preferred instead that the astronauts navigate around gimbal lock (see https://ntrs.nasa.gov/search.jsp?R=19620002325).

In our recent paper,

E.G. Hemingway and Oliver M. O'Reilly "Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and moments" Multibody System Dynamics. (2018).

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 (and thus ineffective in changing the accelerations of the generalized coordinates). Numerous examples, including the classic bead on a rotating hoop example that appears in many dynamics textbooks 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 (as would have happened in the Apollo program were it not for the desire to save weight) and demonstrate how integrable constraints alter the configuration manifold and may consequently eliminate coordinate singularities. The latter phenomenon pertains to the coordinates used to parameterize the configuration manifold of the system, in contrast to gimbal lock which has a distinctive physical manifestation.