Geometry-Driven Stochastic Modeling of SE(3) States Based on Dual Quaternion Representation

We present a novel approach to stochastically model uncertain 6-DoF rigid body motions represented by dual quaternions. Unlike conventional methods relying on the local linearization of the nonlinear SE(3) group, the proposed distribution directly models uncertainty on the manifold of unit dual quaternions. For that, the Bingham distribution is employed on the 3-sphere to model the real part, at which the tangent plane of the hypersphere is spanned by a basis preserving the Bingham principal directions via parallel transport. The conditioning dual part is then expressed with respect to the transported basis and modeled by a Gaussian distribution. This enables the probabilistic interpretation of the correlation between rotation and translation terms. We further introduce the corresponding sampling-approximation scheme for the proposed density, based on which unscented transform-based 6-DoF pose filtering approaches are established and evaluated with simulations.