Gaalop High Performance Computing based on Conformal Geometric Algebra

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1 Gaalop High Performance Computing based on Conformal Geometric Algebra Prof. Andreas Koch, Germany

2 Overview What is Gaalop? Related work Our concepts Proof-of-concept application State-of-the-art and future perspectives

3 Goal of Gaalop? Combination of Elegance of geometric computing based on Geometric Algebra High performance of the implementation

4 Related Work Software solutions Gaigen 2, [Fontijne/Dorst] Hardware solutions Implementation of a Clifford algebra co-processor design on a field programmable gate array, [Perwass et al, 2003] CliffoSor, an Innovative FPGA-based Architecture for Geometric Algebra, [Gentile et al, 2005] Hardware implementation of a geometric algebra processor core, [Mishra/Wilson, 2006]

5 Implementation of a CA co-processor design on a FPGA Perwass et al. 2002

6 Implementation of a CA co-processor design on FPGA Perwass et al. 2002

7 Hardware implementation of a geometric algebra processor core Mishra/Wilson, 2006 Current setup:

8 Current hardware solutions Geometric Algebra Algorithm Hardware solution

9 Gaalop (Concepts) Combination of the advantages of software and hardware optimizations 1. Optimize geometric algebra algorithm symbolically 2. Generate a hardware implementation of the (already symbolically optimized) algorithm

10 Gaalop (Architecture) Geometric Algebra Algorithm Symbolic simplification (Maple) IR (Intermediate( Representation) FPGA C-CodeCode CluCalc Java Latex

11 Symbolic simplification (based on Maple) Goal: Simplifying of expressions to the most efficient ones (addition and multiplications) CliffordLib for Maple [Ablamowicz & Fauser] OpenMaple-Interface

12 The general data structure of Gaalop Focus on 32-D multivector of the conformal model Goal: Compute Symbolically optimized coefficients

13 Example Calculation of the horizon circle:

14 Example Note: different formulas lead to the same result

15 How to implement on hardware? Sdaf Use of parallelism!

16 Proof-of-concept application Competitive runtime performance for inverse kinematics algorithms using conformal geometric algebra Eurographics 2006 Dietmar Hildenbrand, Daniel Fontijne, Yusheng Wang, Marc Alexa, Leo Dorst

17 Development of the inverse kinematics Visual development and test with CLUCalc (free software by Christian Perwass, Easy and intuitive operations with basic geometric entities Entities like spheres and planes Outer product ^ to intersect

18 Symbolic simplification of the inverse kinematics

19 Symbolic simplification of the inverse kinematics

20 Symbolic simplification of the inverse kinematics

21 Symbolic simplification of the inverse kinematics

22 Symbolic simplification of the inverse kinematics

23 Symbolic simplification of the inverse kinematics Maple functions like simplify(), coeff() C_e[9] := -1/2*pwx*L1^2 C_e[10] := pwx C_e[16] :=

24 Symbolic simplification (continued) elbow circle already computed Intersect with swivel plane Resulting point pair Selection of one of the two points leads to:

25 Runtime performance of the inverse kinematics On the algorithmic level Embedding of quaternions ( needed for the slerp motion ) Two optimization approaches for the implementation Gaigen 2 Maple with Cliffordlib -> Three times faster than conventional solution!

26 Paper Efficient inverse kinematics algorithm based on Conformal Geometric Algebra using reconfigurable hardware " VISIGRAPP 2008 Dietmar Hildenbrand Holger Lange Florian Stock Andreas Koch Extention to hardware implementation

27 Way to efficiency? Geometric algebra algorithm Symbolic simplification C-CodeCode reconfigurable hardware FPGA

28 HW implementation Dataflow graph for with

29 HW implementation Pipeline stages

30 Parallelism in 2 dimensions 363 Pipeline stages, up to 20 scalar operations per stage

31 HW implementation results 100 inverse kinematics calculation Test platform FPGA: Xilinx Virtex XC2VP70-7, 100 MHz PC: Intel Centrino M7115CPU, 1.5 GHz

32 Results summary Maple-optimized implementation about 3 times faster than the conventional solution Currently with reconfigurable hardware: more than 500 times faster than the conventional solution

33 Gaalop state-of-the-art Geometric Algebra Algorithm (Clucalc, Maple) Symbolic simplification (Maple) IR (Intermediate( Representation) FPGA C-CodeCode CluCalc Java Latex

34 Gaalop future perspectives Geometric Algebra Algorithm Symbolic simplification (Maple) IR (Intermediate( Representation) FPGA C-CodeCode CluCalc Java Latex

35 Benefits of Geometric Computing with Gaalop Geometrically intuitive Easy to understand Easy to develop Easy to maintain and Efficient to implement Software Hardware

36 Acknowlegments Andreas Koch, Embedded Systems Group, TU Darmstadt Eduardo Bayro Joachim Pitt

37 Thank you for your attention

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