Geometric Algebra for Optimal Control with Applications in Manipulation Tasks

Many problems in robotics are fundamentally problems of geometry, which lead to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie algebra and dual quaternions. A unification and generalization of these popular formalisms can be found in geometric algebra. The aim of this paper is to showcase the capabilities of geometric algebra when applied to robot manipulation tasks. In particular the modelling of cost functions for optimal control can be done uniformly across different geometric primitives leading to a low symbolic complexity of the resulting expressions and a geometric intuitiveness. We demonstrate the usefulness, simplicity and computational efficiency of geometric algebra in several experiments using a Franka Emika robot. The presented algorithms were implemented in c++20 and resulted in the publicly available library \textit{gafro}. The benchmark shows faster computation of the kinematics than state-of-the-art robotics libraries.Comment: 16 pages, 13 figures

Computation with Curved Shapes: Towards Freeform Shape Generation in Design

Shape computations are a formal representation that specify particular aspects of the design process with reference to form. They are defined according to shape grammars, where manipulations of pictorial representations of designs are formalised by shapes and rules applied to those shapes. They have frequently been applied in architecture in order to formalise the stylistic properties of a given corpus of designs, and also to generate new designs within those styles. However, applications in more general design fields have been limited. This is largely due to the initial definitions of the shape grammar formalism which are restricted to rectilinear shapes composed of lines, planes or solids. In architecture such shapes are common but in many design fields, for example industrial design, shapes of a more freeform nature are prevalent. Accordingly, the research described in this thesis is concerned with extending the applicability of the shape grammar formalism such that it enables computation with freeform shapes. Shape computations utilise rules in order to manipulate subshapes of a design within formal algebras. These algebras are specified according to embedding properties and have previously been defined for rectilinear shapes. In this thesis the embedding properties of freeform shapes are explored and the algebras are extended in order to formalise computations with such shapes. Based on these algebras, shape operations are specified and algorithms are introduced that enable the application of rules to shapes composed of freeform B´ezier curves. Implementation of the algorithms enables the application of shape grammars to shapes of a more freeform nature than was previously possible. Within this thesis shape grammar implementations are introduced in order to explore both theoretical issues that arise when considering computation with freeform shapes and practical issues concerning the application of shape computation as a model for design and as a mode for generating freeform shapes

Representation Theory of Chern Simons Observables

Recently we suggested a new quantum algebra, the moduli algebra, which was conjectured to be a quantum algebra of observables of the Hamiltonian Chern Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.Comment: 63 pages, late

Functional maps representation on product manifolds

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices

Functional Maps Representation on Product Manifolds

We consider the tasks of representing, analyzing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace--Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru