Deterministic Attitude and Pose Filtering, an Embedded Lie Groups Approach

Attitude estimation is a core problem in many robotic systems that perform automated or semi automated navigation. The configuration space of the attitude motion is naturally modelled on the Lie group of special orthogonal matrices SO(3). Many current attitude estimation methods are based on non-matrix parameterization of attitude. Non-matrix parameterization schemes sometimes lead to modelling issues such as the singularities in the parameterization space, non-uniqueness of the attitude estimates and the undesired conversion errors such as the projection or normalization errors. Moreover, often attitude filters are designed by linearizing or approximating the nonlinear attitude kinematics followed by applying the Kalman filtering based methods that are primarily only suitable for linear Gaussian systems. In this thesis, the attitude estimation problem is considered directly on SO(3) along with nonlinear vectorial measurement models. Minimum-energy filtering is adapted to respect the geometry of the problem and in order to solve the problem avoiding linearization or Gaussian assumptions. This approach allows for obtaining a geometric approximate minimum-energy (GAME) filter whose performance is tested by means of Monte Carlo simulations. Many of the major attitude filtering methods in the literature are surveyed and included in the simulation study. The GAME filter outperforms all of the state of the art attitude filters studied, including the multiplicative extended Kalman filter (MEKF), the unscented quaternion estimator (USQUE), the right-invariant extended Kalman filter (RIEKF) and the nonlinear constant gain attitude observer, in the asymptotic estimation error. Furthermore, the proposed GAME filter is shown to be near-optimal by deriving a bound on the optimality error of the filter that is proven to be small in simulations. Moreover, similar GAME filters are derived for pose filtering on the special Euclidean group SE(3), attitude and bias filtering on the unit circle and attitude and bias filtering on the special orthogonal group. The approximation order of the proposed method can potentially be extended to arbitrary higher orders. For instance, for the case angle estimation on the unit circle an eighth-order approximate minimum-energy filter is provided.