Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds

Common estimation algorithms, such as least squares estimation or the Kalman filter, operate on a state in a state space S that is represented as a real-valued vector. However, for many quantities, most notably orientations in 3D, S is not a vector space, but a so-called manifold, i.e. it behaves like a vector space locally but has a more complex global topological structure. For integrating these quantities, several ad-hoc approaches have been proposed. 
Here, we present a principled solution to this problem where the structure of the 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 local vector-space view \delta; --> x [+] \delta; around a given state x. Generic estimation algorithms can then work on the manifold S mainly by replacing +/- with [+]/[-] where appropriate. We analyze these operators axiomatically, and demonstrate their use in least-squares estimation and the Unscented Kalman Filter. Moreover, we exploit the idea of encapsulation from a software engineering perspective in the Manifold Toolkit, where the [+]/[-] operators mediate 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

The need to evaluate expressions of the form $f(A)$ or $f(A)b$, where $f$ is a nonlinear function, $A$ is a large sparse $n\times n$ matrix, and $b$ is an $n$-vector, arises in many applications This paper describes how the Faber transform applied to the field of values of $A$ can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed

/pdf/error-estimates-and-evaluation-of-matrix-functions-via-the-3bu7zkr56l.pdf

Error Estimates and Evaluation of Matrix Functions via the Faber Transform

In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge–Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Stormer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form.

In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.

Hamilton–Pontryagin Integrators on Lie Groups Part I: Introduction and Structure-Preserving Properties

Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing geodesic flows in the Hamilton–Jacobi representation. This paper takes the differential geometric basis of Markov chain Monte Carlo further by considering methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics. Proposal mechanisms are developed based on the geodesic flows over the manifolds of support for the distributions, and illustrative examples are provided for the hypersphere and Stiefel manifold of orthonormal matrices.

/pdf/geodesic-monte-carlo-on-embedded-manifolds-cr6cwznomx.pdf

Geodesic Monte Carlo on Embedded Manifolds

Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

https://estudogeral.sib.uc.pt/bitstream/10316/11463/1/The%20Moser-Veselov%20equation.pdf

The Moser-Veselov equation

The exponential‐mean‐log‐transference as a possible representation of the optical character of an average eye

https://estudogeral.sib.uc.pt/bitstream/10316/4648/1/filecc3307f1fa2640c4bcb8705ac3c5e96c.pdf

João R. Cardoso

Papers

Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

The Moser-Veselov equation

The exponential‐mean‐log‐transference as a possible representation of the optical character of an average eye

Theoretical and numerical considerations about Padé approximants for the matrix logarithm

Some notes on properties of the matrix Mittag-Leffler function