Christian Lageman

Gradient-Like Observers for Invariant Dynamics on a Lie Group

This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialized to systems on Lie groups. For invariant systems this leads to a general factorization theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.

Nonlinear attitude observers on SO(3) for complementary and compatible measurements: A theoretical study

This paper considers the question of designing an attitude observer exploiting the structure of the Special Orthogonal Group SO(3) for both inertial and body-fixedframe measurements. We consider measurements from a minimal sensor suite, typically a rate gyroscope along with several measurements of inertial and/or body-fixed vector directions. We propose fully nonlinear pose observers based directly on the vector measurements, allowing a mix of both inertial and body-fixed measurements.We provide a comprehensive stability analysis of the observer error dynamics. We show that even with a single vector measurement it is often possible to recover asymptotically stable observer error dynamics.

Analysis of Non-Linear Attitude Observers for Time-Varying Reference Measurements

This paper provides a comprehensive stability analysis of a suite of nonlinear attitude observers that have been developed over the last few years. The observers considered are based on vectorial measurements of an a priori known reference direction. By treating the reference direction and the measurement in the same analysis framework, and allowing time-variation of either, we are able to define general persistency of excitation criteria that incorporate and generalize convergence criteria used in prior work. A key outcome is conditions that ensure almost global asymptotic and local exponential stability of attitude observers based on a single vector measurement as long as the excitation conditions are met on the reference and system trajectory. The approach generalizes stability results provided in prior work, based on rank conditions, that required at least two or more vector measurements.

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.

Synchronization with partial state coupling on SO(n)

This paper studies autonomous synchronization of

Decompounding on Compact Lie Groups

k

Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

agent orientations

Bias estimation for invariant systems on Lie groups with homogeneous outputs

\in SO(n)

Newton-type algorithms for time-varying pose estimation

by using links that each convey (a priori different) partial state information between two agents: the latter compare the action of their states on a common link-dependent vector of

Synchronization with partial state feedback on SO(n)

\mathbb{R}^n