GAALOPWeb for Matlab: An Easy to Handle Solution for Industrial Geometric Algebra Implementations
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Advances in Applied Clifford Algebras
GAALOPWeb for Matlab: An Easy to Handle Solution for Industrial Geometric Algebra Implementations Dietmar Hildenbrand, Christian Steinmetz and Radek Tich´ y∗ Abstract. We present GAALOPWeb for Matlab, a new easy to handle solution for Geometric Algebra implementations for Matlab. We demonstrate its usability for industrial applications based on a forward kinematics algorithm of a serial robot arm and illustrate it with the help of high run-time performance. Mathematics Subject Classification. Primary 15A66 . Keywords. Conformal Geometric Algebra, Kinematics, Code optimization.
1. Introduction The origin of Geometric Algebra computations goes back to 1966 when Hestenes introduced this calculus for physicists in [4]. Since then, the contribution expanded to many engineering areas, such as computer graphics, computer vision and mechanics, see e.g. [14]. Geometric algebra Gn+1,1 of conformal space is Clifford algebra Cl(n + 1, 1) of quadratic space Rn+1,1 which provides a model for conformal geometries of Euclidean space. The conformal model of 3D Euclidean space known as Conformal Geometric Algebra1 G4,1 (CGA) and its applications were studied, for instance, in [3,10]. With the help of this tool it is very easy to deal with Euclidean transformations of geometric primitives and as a consequence both development and possibilities for applications expanded. The third author was supported by a Grant Number FSI-S-17-4464. This article is part of the ENGAGE 2019 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Linwang Yuan (EiC), Werner Benger, Dietmar Hildenbrand, and Eckhard Hitzer. ∗ Corresponding 1 its
author. modification compatible with conic transformation, for instance, was introduced in [9] 0123456789().: V,-vol
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D. Hildenbrand, C. Steinmetz
Adv. Appl. Clifford Algebras
In this paper, we focus on the investigation of CGA computations based on Matlab which is very popular in the engineering and research community. For this purpose we originally used the Clifford Multivector Toolbox for Matlab [13] in order to develop a kinematics algorithm of a real robot. This toolbox2 provides a very intuitive way how Matlab users can handle computations in Geometric Algebra. But, we realized that the runtime performance was far from being acceptable for industrial applications. On the other hand there was the GAALOP software as introduced in the books [5] and [6] providing optimized solutions for programming languages such as C/C++. The goal of this paper is to close this gap and provide a tool for Matlab users which is • easy to access, • easy to use, • providing a high runtime performance for industrial applications. Section 2 shows the high expressiveness of Conformal Geometric Algebra (CGA) as a mathematical language. As a sample manipulator we use the robot ABB IRB4400 and for verification of the algorithm functionality we use ABB RobotStudio. The manipulator model is described in Sect. 3. Its kinematics3 is demonstrated in Sect. 4
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