Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds

Common estimation algorithms, such as least squares estimation or the Kalman filter, operate on a state in a state space S that is represented as a real-valued vector. However, for many quantities, most notably orientations in 3D, S is not a vector space, but a so-called manifold, i.e. it behaves like a vector space locally but has a more complex global topological structure. For integrating these quantities, several ad hoc approaches have been proposed. Here, we present a principled solution to this problem where the structure of the manifold S is encapsulated by two operators, state displacement :S×Rn→S and its inverse :S×S→Rn. These operators provide a local vector-space view δ ↦ x δ around a given state x. Generic estimation algorithms can then work on the manifold S mainly by replacing +/− with / where appropriate. We analyze these operators axiomatically, and demonstrate their use in least-squares estimation and the Unscented Kalman Filter. Moreover, we exploit the idea of encapsulation from a software engineering perspective in the Manifold Toolkit, where the / operators mediate between a "flat-vector" view for the generic algorithm and a "named-members" view for the problem specific functions.