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Constrained Dynamics in Conformal and Projective Geometric Algebra

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Advances in Computer Graphics (CGI 2020)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 12221))

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In this paper we tackle the problem of constrained rigid body dynamics in the Conformal and Projective Geometric Algebras (CGA, PGA). First we construct a screw-theory based formulation of dynamics in CGA and note the equivalence between this and the PGA dynamics presented by Gunn in [1]. After verifying the formulation via simulation, we move on to the challenge of adding constraints. First we apply the standard mechanical engineering technique of virtual power to the constraint problem in our Geometric Algebra (GA) framework. We then discuss a novel technique for ‘pinning’ dynamic rigid bodies to geometric primitives, a technique that relies on the invariance of certain multivectors and functions of multivectors to specific rotor transformations.

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References

  1. Gunn, C.: On the homogeneous model of Euclidean geometry. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice, pp. 297–327. Springer, London (2011). https://doi.org/10.1007/978-0-85729-811-9_15

    Chapter  Google Scholar 

  2. Pottmann, H., Wallner, J.: Computational Line Geometry. MATHVISUAL. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-642-04018-4

    Book  MATH  Google Scholar 

  3. Lasenby, A., Lasenby, R., Doran, C.: Rigid body dynamics and conformal geometric algebra. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice, pp. 3–24. Springer, London (2011). https://doi.org/10.1007/978-0-85729-811-9_1

    Chapter  MATH  Google Scholar 

  4. Candy, L.P.: Kinematics in conformal geometric algebra with applications in strapdown inertial navigation. Ph.D. thesis. University of Cambridge (2012)

    Google Scholar 

  5. Boyle, M.: The integration of angular velocity. Adv. Appl. Clifford Algebras 27(3), 2345–2374 (2017). https://doi.org/10.1007/s00006-017-0793-z

    Article  MathSciNet  MATH  Google Scholar 

  6. Gunn, C.: Non-metric alternatives to reciprocal frames. The bivector.net forums. https://discourse.bivector.net/t/non-metric-alternative-to-reciprocal-frame/105/4. Accessed 12 May 2020

  7. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry, 1st edn. Elsevier, Morgan Kaufmann, Amsterdam (2007)

    Google Scholar 

  8. Gallardo-Alvarado, J.: Kinematic Analysis of Parallel Manipulators by Algebraic Screw Theory. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31126-5

    Book  MATH  Google Scholar 

  9. Arsenovic, A., Hadfield, H., Wieser, E., Kern, R., The Pygae Team: Clifford (Version v1.3.0), 29 May 2020. Zenodo. https://doi.org/10.5281/zenodo.3865446

  10. Du, J., Goldman, R., Mann, S.: Modeling 3D geometry in the Clifford algebra R(4, 4). Adv. Appl. Clifford Algebras 27(4), 3039–3062 (2017). https://doi.org/10.1007/s00006-017-0798-7

    Article  MathSciNet  MATH  Google Scholar 

  11. Easter, R.B., Hitzer, E.: Double conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2175–2199 (2017). https://doi.org/10.1007/s00006-017-0784-0

    Article  MathSciNet  MATH  Google Scholar 

  12. Breuils, S., Nozick, V., Sugimoto, A., Hitzer, E.: Quadric conformal geometric algebra of \({\mathbb{R}}^{9,6}\). Adv. Appl. Clifford Algebras 28(2), 1–16 (2018). https://doi.org/10.1007/s00006-018-0851-1

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Cambridge, UK

    Hugo Hadfield & Joan Lasenby

Authors
  1. Hugo Hadfield
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  2. Joan Lasenby
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Corresponding author

Correspondence to Hugo Hadfield .

Editor information

Editors and Affiliations

  1. University of Geneva, Geneva, Switzerland

    Nadia Magnenat-Thalmann

  2. University of Crete, Heraklion, Greece

    Constantine Stephanidis

  3. University of Macau, Macau, China

    Enhua Wu

  4. Swiss Federal Institute of Technology, Lausanne, Switzerland

    Daniel Thalmann

  5. Shanghai Jiao Tong University, Shanghai, China

    Bin Sheng

  6. University of Sydney, Sydney, Australia

    Jinman Kim

  7. University of Crete, Heraklion, Greece

    George Papagiannakis

  8. University of Calgary, Calgary, AB, Canada

    Marina Gavrilova

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Hadfield, H., Lasenby, J. (2020). Constrained Dynamics in Conformal and Projective Geometric Algebra. In: Magnenat-Thalmann, N., et al. Advances in Computer Graphics. CGI 2020. Lecture Notes in Computer Science(), vol 12221. Springer, Cham. https://doi.org/10.1007/978-3-030-61864-3_39

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  • DOI: https://doi.org/10.1007/978-3-030-61864-3_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-61863-6

  • Online ISBN: 978-3-030-61864-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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