Observers for invariant systems on Lie groups with biased input measurements and homogeneous outputs

This paper provides a new observer design methodology for invariant systems whose state evolves on a Lie group with outputs in a collection of related homogeneous spaces and where the measurement of system input is corrupted by an unknown constant bias. The key contribution of the paper is to study the combined state and input bias estimation problem in the general setting of Lie groups, a question for which only case studies of specific Lie groups are currently available. We show that any candidate observer (with the same state space dimension as the observed system) results in non-autonomous error dynamics, except in the trivial case where the Lie-group is Abelian. This precludes the application of the standard non-linear observer design methodologies available in the literature and leads us to propose a new design methodology based on employing invariant cost functions and general gain mappings. We provide a rigorous and general stability analysis for the case where the underlying Lie group allows a faithful matrix representation. We demonstrate our theory in the example of rigid body pose estimation and show that the proposed approach unifies two competing pose observers published in prior literature.Comment: 11 page

Local observers on linear Lie groups with linear estimation error dynamics

This paper proposes local exponential observers for systems on linear Lie groups. We study two different classes of systems. In the first class, the full state of the system evolves on a linear Lie group and is available for measurement. In the second class, only part of the system's state evolves on a linear Lie group and this portion of the state is available for measurement. In each case, we propose two different observer designs. We show that, depending on the observer chosen, local exponential stability of one of the two observation error dynamics, left- or right-invariant error dynamics, is obtained. For the first class of systems these results are developed by showing that the estimation error dynamics are differentially equivalent to a stable linear differential equation on a vector space. For the second class of system, the estimation error dynamics are almost linear. We illustrate these observer designs on an attitude estimation problem

State Estimation for Systems on Lie Groups with Nonideal Measurements

This thesis considers the state estimation problem for invariant systems on Lie groups with inputs in its associated Lie algebra and outputs in homogeneous spaces of the Lie group. A particular focus of this thesis is the development of state estimation methodologies for systems with nonideal measurements, especially systems with additive input measurement bias, output measurement delay, and sampled outputs. The main contribution of the thesis is to effectively employ the symmetries of the system dynamics and to benefit from the Lie group structure of the underlying state space in order to design robust state estimators that are computationally simple and are ideal for embedded applications in robotic systems. We address the input measurement bias problem by proposing a novel nonlinear observer to adaptively eliminate the input measurement bias. Despite the nonlinear and non-autonomous nature of the resulting error dynamics and the complexity of the underlying state space, the proposed observer exhibits asymptotic/exponential convergence of the state and bias estimation errors to zero. To tackle the output measurement delay problem, we propose novel dynamic predictors used in an observer-predictor arrangement. The observer provides estimates of the delayed state using the delayed output measurements and the predictor takes those estimates, compensates for the delay, and provides predictions of the current state. Separately, we propose output predictors employed in a predictor-observer arrangement to address the problem of sampled output measurements. The output predictors take the sampled measurements and provide continuous predictions of the current outputs. Feeding the predicted outputs into the observer yields estimates of the current state. Both methods rely on the invariance of the underlying system dynamics to recursively provide predictions with low computation requirements. We demonstrate applications of the theory with examples of attitude, velocity, and position estimation on SO(3) and SE(3). A key contribution of this thesis is the development of C++ libraries in an embedded implementation as well as experimental verification of the developed theory with real flight tests using model UAVs