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.

💡 Research Summary

The paper addresses the challenge of estimating a unit‑bearing vector in image‑based visual servoing (IBVS) when both angular velocity (from an IMU) and linear velocity (derived from optical flow) affect the target’s apparent motion. The bearing dynamics are modeled on the 2‑sphere S² as ˙b = –S(ω)b + v̅, where ω∈ℝ³ is the measured angular velocity and v̅∈ℝ³ is a distance‑normalized linear velocity that lies in the tangent space of the sphere at b. While previous works considered only the angular term, this study incorporates the linear term to capture the vehicle’s translational motion, which is essential for realistic IBVS applications.

To exploit the geometric symmetries of the problem, the authors lift the dynamics to the Lie group SO(3). They define a right group action ϕ:SO(3)×S²→S² by ϕ(X,ξ)=Xᵀξ and an input transformation ψ:SO(3)×(ℝ³×ℝ³)→(ℝ³×ℝ³) by ψ(X,(ω,v̅))=(Xᵀω, Xᵀv̅). These definitions satisfy the equivariance condition Dϕ_X