Frenet oscillations and Frenet–Euler angles: curvature singularity and motion-trajectory analysis

Motion-trajectory (MT) curves are used to introduce Frenet oscillations. Time-varying orientation of the motion plane that contains absolute velocity and acceleration vectors is defined in terms three Frenet–Euler angles; curvature, vertical-development, bank angles, referred as angles for brevity. The angle associated super-elevation plane, which measure deviation centrifugal inertia force from horizontal can be shed light on definition balance speed practice. concept pre-super-elevated osculating (PSEO) introduced Rodrigues’ formula employed develop an orthogonal rotation matrix provides a geometric interpretation PSEO plane. A new inverse-dynamics problem utilizes experimentally or simulation recorded trajectories (RMT) define forces demonstrate their equivalence Cartesian form forces. New expressions curvature vector acceleration, limit magnitude tangential given forward velocity, condition required remain horizontal, singular points derived. prove existence normal at points. It shown assume different forms, depending curve parameter used. results simple analytical oscillations importance distinguishing between highway-ramp railroad track super-elevations, time-invariant, super-elevation, motion-dependent.