Embedded Coprocessors for Native Execution of Geometric Algebra Operations

• Published in: Advances in applied Clifford algebras Vol. 27; no. 1; pp. 559 - 580
• Main Authors: Franchini, Silvia, Gentile, Antonio, Sorbello, Filippo, Vassallo, Giorgio, Vitabile, Salvatore
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2017; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Coprocessors; Field programmable gate arrays; Gate arrays; Hardware; Image processing; Inverse kinematics; Kinematics; Mathematical and Computational Physics; Mathematical Methods in Physics; Microprocessors; Motion capture; Operators (mathematics); Physics; Physics and Astronomy; Programmable logic controllers; Representations; Robot dynamics; Theoretical; Transformations (mathematics); Vector spaces; 68W30; Primary 15A66; Geometric algebra; Embedded coprocessors; FPGA-based prototyping; Application-specific processors; Secondary 68U05
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-016-0662-1

Clifford algebra or geometric algebra (GA) is a simple and intuitive way to model geometric objects and their transformations. Operating in high-dimensional vector spaces with significant computational costs, the practical use of GA requires dedicated software and/or hardware architectures to directly support Clifford data types and operators. In this paper, a family of embedded coprocessors for the native execution of GA operations is presented. The paper shows the evolution of the coprocessor family focusing on the latest two architectures that offer direct hardware support to up to five-dimensional Clifford operations. The proposed coprocessors exploit hardware-oriented representations of GA elements and operators properly conceived to obtain fast performing implementations. The coprocessor prototypes, implemented on field programmable gate arrays development boards, show significant speedups of about one order of magnitude with respect to the baseline software library Gaigen running on a general-purpose processor. The paper also presents an execution analysis of different GA-based applications, namely inverse kinematics of a robot, optical motion capture, raytracing, and medical image processing, showing good speedups with respect to the baseline general-purpose implementation.