Variation Based Extended Kalman Filter on $S^{2}$

In this paper, we propose a variation-based extended Kalman filter (V-EKF) on the two-sphere manifold. We consider the spherical pendulum dynamical system whose nonlinear geometric dynamics evolve on the two-sphere. These dynamics are linearized about the current state using a variation-based linearization resulting in a time-varying linear system with state constraints that describe the dynamics of the variation states. The Kalman filter is applied on the resulting variation states with the pendulum position as measurements for measurement updates. The V-EKF also has a constraint update where the estimated state and covariance are updated to ensure they satisfy the constraints. Desirable properties of V-EKF, such as preserving the geometric structure of the estimated state are thus achieved. The proposed method is illustrated through numerical simulations and also validated through experiments.