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Common estimation algorithms, such as least squares estimation or the Kalmanfilter, operate on a state in a state space S that is represented as areal-valued vector. However, for many quantities, most notably orientations in3D, S is not a vector space, but a so-called manifold, i.e. it behaves like avector space locally but has a more complex global topological structure. Forintegrating these quantities, several ad-hoc approaches have been proposed.  Here, we present a principled solution to this problem where the structure ofthe manifold S is encapsulated by two operators, state displacement [+]:S x R^n--> S and its inverse [-]: S x S --> R^n. These operators provide a localvector-space view \delta; --> x [+] \delta; around a given state x. Genericestimation algorithms can then work on the manifold S mainly by replacing +/-with [+]/[-] where appropriate. We analyze these operators axiomatically, anddemonstrate their use in least-squares estimation and the Unscented KalmanFilter. Moreover, we exploit the idea of encapsulation from a softwareengineering perspective in the Manifold Toolkit, where the [+]/[-] operatorsmediate between a "flat-vector" view for the generic algorithm and a"named-members" view for the problem specific functions.

Integrating Generic Sensor Fusion Algorithms with Sound State Representations through Encapsulation of Manifolds