Symbolic and User-friendly Geometric Algebra Routines (SUGAR) for Computations in Matlab

2024-03-25

ACM Transactions on Mathematical Software

AI Summary

Geometric algebra (GA) has long been hindered in applied mathematics and engineering practice due to the lack of symbolic implementation and engineering-friendly interfaces. Method: This paper introduces SUGAR, an open-source MATLAB toolbox that uniquely integrates symbolic derivation with numerical computation for projective and conformal GA in 2D/3D. Built upon MATLAB’s Symbolic Math Toolbox, it implements a custom multivector class with operator overloading, rotor generation and interpolation, conformal mapping, and automatic dimension adaptation, supporting unified evaluation of multivector functions (e.g., exp, log, sin, cos) and rotor-driven transformations. Contribution/Results: Evaluated on robotic kinematics modeling, robust controller design, and three-phase power system simulation, SUGAR significantly simplifies modeling workflows, enhances computational interpretability and code reusability, and bridges the gap toward practical GA adoption in engineering applications.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

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 paper introduces SUGAR (Symbolic and User-friendly Geometric Algebra Routines), 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 paper also presents three validation examples across these fields, showcasing SUGAR’s practical utility in solving real-world engineering and applied mathematics problems.

Problem

• Addresses scarcity of symbolic geometric algebra tools for engineers
• Simplifies complex geometric algebra computations in Matlab
• Enhances GA accessibility for applied mathematics and engineering

Innovation

• Open-source Matlab toolbox for geometric algebra
• Supports symbolic and numeric GA computations
• Tailored for engineering and applied mathematics