Equivariant Observer for Bearing Estimation with Linear and Angular Velocity Inputs

This work addresses the problem of designing an equivariant observer for a first order dynamical system on the unit-sphere. Building upon the established case of unit bearing vector dynamics with angular velocity inputs, we introduce an additional linear velocity input projected onto the unit-sphere tangent space. This extended formulation is particularly useful in image-based visual servoing scenarios where stable bearing estimates are required and the relative velocity between the vehicle and target features must be accounted for. Leveraging lifted kinematics to the Special Orthogonal group, we design an observer for the bearing vector and prove its almost global asymptotic stability. Additionally, we demonstrate how the equivariant observer can be expressed in the original state manifold. Numerical simulation results validate the effectiveness of the proposed algorithm.