Symbolic and user-friendly geometric algebra routines (SUGAR) for computations in Matlab

Abstract

Geometric Algebra (GA) provides a unified, compact mathematical framework for geometric computing, simplifying relations typically handled with more complex tools like matrix multiplication. In fields like robotics, GA replaces conventional coordinate-based approaches with the multiplication of special elements called rotors, offering greater efficiency. Despite its advantages, GA’s complexity and the lack of symbolic tools hinder its broader adoption among applied mathematicians and engineers. To address this, this article introduces Symbolic and User-friendly Geometric Algebra Routines (SUGAR), an open source Matlab toolbox. SUGAR streamlines GA usage in Matlab through a collection of user-friendly functions that support both numeric and symbolic computations, even in high-dimensional algebras. Designed for applied mathematics and engineering, it enables intuitive manipulation of geometric elements and transformations in two- and three-dimensional projective and conformal GAs, consistent with established computational methods. Moreover, SUGAR manages multivector functions such as exponential, logarithmic, sinusoidal, and cosine operations, enhancing its applicability in domains like robotics, control systems, and power electronics. Finally, this article also presents three validation examples across these fields, showcasing SUGAR’s practical utility in solving real-world engineering and applied mathematics problems.

The work of I. Zaplana was partially supported by the Spanish National Project PID2020-114819GB-I00 funded by MI- CIU/AEI/10.13039/501100011033 and by the Generalitat de Catalunya through the Project 2021 SGR 00375. The work of A. Dòria-Cerezo was partially supported by the Spanish National Project PID2021-122821NB-I00 funded by MI- CIU/AEI/10.13039/501100011033 and ERDF/EU and by the Generalitat de Catalunya through the Project 2021 SGR 00376. M. Velasco, J. Duarte, and P. Martí were partially supported by the Spanish National Project PID2021-122835OB-C21 funded by MICIU/AEI/10.13039/501100011033.

Peer Reviewed

Postprint (author's final draft)