A spherical amplitude-phase formulation for 3-D adaptive line-of-sight (ALOS) guidance with USGES stability guarantees

A recently proposed 3-D adaptive line-of-sight (ALOS) path-following algorithm addressed coupled motion dynamics of marine craft, aircraft, and uncrewed vehicles under environmental disturbances such as wind, waves, and ocean currents. Stability analysis established uniform semiglobal exponential stability (USGES) of the cross- and vertical-track errors using a body-velocity-based amplitude-phase representation of the North-East-Down (NED) kinematic differential equations. In this brief paper, we revisit the ALOS framework and introduce a novel spherical amplitude-phase representation. This formulation yields a more geometrically intuitive and physically observable description of the guidance errors and enables a significantly simplified stability proof. Unlike the previous model, which relied on a vertical crab angle derived from body-frame velocities, the new representation uses an alternative vertical crab angle and retains the USGES property. It also removes restrictive assumptions such as constant altitude/depth or zero horizontal crab angle, and remains valid for general 3-D maneuvers with nonzero roll, pitch, and flight-path angles.