Application of the manifold-constrained unscented Kalman filter

This document describes the rationale and methodology behind the application of a Kalman-type filter to a system that has two properties which lead to inaccuracy or instability in traditional filters: highly non-linear system models along with a state that is constrained to a non-linear Riemannian manifold. The non-linear models are handled by the use of the unscented transformation, while the constrained state is dealt with using both a modified unscented transformation and a modified time-update model. The application that requires these treatments is the system identification of a super-light unmanned aerial vehicle, where the dynamics of the vehicle are such that an unconstrained orientation must be dealt with as a unit-quaternion, the high-order of the model requires maximum precision be maintained, and the vehicle itself requires the lowest-mass sensors available, leading to relatively high sensor noise in an already noisy measurement environment. The new filter is explained in this context, implementation details are given, and results of simulation and flight trials are explored. In addition, square-root extensions to this filter are described that increase the filter's computational efficiency without sacrificing its accuracy, stability, or robustness.