12,749 research outputs found
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
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
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