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.

翻译：机器人领域的许多问题本质上是几何问题，这促使近年来对机器人学几何方法的研究力度不断加大。相关成果产生了利用螺旋理论、李代数和对偶四元数等不同框架的算法。几何代数能够统一并推广这些流行形式体系。本文旨在展示几何代数应用于机器人操作任务时的能力，特别是在最优控制的代价函数建模中，该方法可统一处理不同几何基元，从而降低所得表达式的符号复杂度并增强几何直观性。我们通过使用Franka Emika机器人的多项实验，验证了几何代数的实用性、简洁性与计算效率。所提出的算法基于C++20实现，并形成了开源库\textit{gafro}。基准测试表明，其运动学计算速度优于当前最先进的机器人学库。