Publication

Geometric Algebra for Optimal Control With Applications in Manipulation Tasks

Résumé

Many problems in robotics are fundamentally problems of geometry, which have led 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 article is to showcase the capabilities of geometric algebra when applied to robot manipulation tasks. In particular, the modeling 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 gafro. The benchmark shows faster computation of the kinematics than state-of-the-art robotics libraries.

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Concepts associés (35)
Algèbre géométrique (structure)
Une algèbre géométrique est, en mathématiques, une structure algébrique, similaire à une algèbre de Clifford réelle, mais dotée d'une interprétation géométrique mise au point par David Hestenes, reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilisé dans un sens plus général pour décrire l'étude et l'application de ces algèbres : l'algèbre géométrique est l'étude des algèbres géométriques).
History of algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property).
Robotique
thumb|upright=1.5|Nao, un robot humanoïde. thumb|upright=1.5|Des robots industriels au travail dans une usine. La robotique est l'ensemble des techniques permettant la conception et la réalisation de machines automatiques ou de robots. L'ATILF donne la définition suivante du robot : « il effectue, grâce à un système de commande automatique à base de microprocesseur, une tâche précise pour laquelle il a été conçu dans le domaine industriel, scientifique, militaire ou domestique ».
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