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 Proofofconcept application Stateoftheart 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 coprocessor design on a field programmable gate array, [Perwass et al, 2003] CliffoSor, an Innovative FPGAbased Architecture for Geometric Algebra, [Gentile et al, 2005] Hardware implementation of a geometric algebra processor core, [Mishra/Wilson, 2006]
5 Implementation of a CA coprocessor design on a FPGA Perwass et al. 2002
6 Implementation of a CA coprocessor 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 CCodeCode 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] OpenMapleInterface
12 The general data structure of Gaalop Focus on 32D 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 Proofofconcept 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 CCodeCode 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 XC2VP707, 100 MHz PC: Intel Centrino M7115CPU, 1.5 GHz
32 Results summary Mapleoptimized implementation about 3 times faster than the conventional solution Currently with reconfigurable hardware: more than 500 times faster than the conventional solution
33 Gaalop stateoftheart Geometric Algebra Algorithm (Clucalc, Maple) Symbolic simplification (Maple) IR (Intermediate( Representation) FPGA CCodeCode CluCalc Java Latex
34 Gaalop future perspectives Geometric Algebra Algorithm Symbolic simplification (Maple) IR (Intermediate( Representation) FPGA CCodeCode 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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