Euler angle gimbal lock elimination with xy+ convention

Abstract

Autonomous satellite navigation theory includes mathematical proofs showing Euler angles developed with current methods for orientation parameterization cannot be mutually orthogonal, since single sequences of three rotations around moving coordinate axes inevitably results in non-orthogonal coordinate axes. A topological proof is embodied in the so–called hairy ball theorem that uses four–dimensional parameterizations (quaternions or angle–axis representations) to demonstrate inability to assign three independent, continuous angular parameters. Another proof illustrates mathematical breakdown when using three sequential elemental coordinate frame rotations where a rotation axis aligns one of the other two axes. At this point called gimbal lock, one degree of freedom is lost, and the mathematical matrix representation drops from rank three to rank two resulting in singularities. A new method eliminating gimbal lock singularities is proposed by parameterizing orientation using three (as opposed to one) sequences of three consecutive rotations leading to the elaboration of three mutually orthogonal Euler angles representing orientation angular rotations about body–fixed axes (not inertial or intermediate axes). Following heuristic and analytic development, validation is provided by space experiments flown in May 2026 using error plots of the difference between classical and proposed mutually orthogonal Euler angles. IMU/INS drift is seen using classical Euler angles, but not with the proposed Euler angles.