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Appendix E: Useful Time Derivatives

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Part of the Simulation Foundations, Methods and Applications book series (SFMA)

Abstract

This Appendix presents a detailed description of how the time derivatives of a normal vector, a rotation matrix and a quaternion are computed. These time derivatives are extensively used in Chaps. 4 and 5 to describe the dynamics of a rigid-body system.

Keywords

  • Derivative Action Time
  • Rotation Matrix Representation
  • Contact Normal Direction
  • Canonical Coordinate Frame
  • Vertex-face Contact

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    In the rigid-body case, the coordinate frame \(\mathcal{F}_{1}\) is the body frame.

  2. 2.

    To simplify the notation, we shall use \(\dot{a}\) to represent the time derivative \(\frac {d a}{dt}\).

References

  1. Baraff, D., Witkin, A.: Physically based modeling. SIGGRAPH Course Notes 13 (1998)

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  2. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press, San Diego (1980)

    MATH  Google Scholar 

  3. Mirtich, B.V.: Impulse-based dynamic simulation of rigid body systems. PhD Thesis, University of California, Berkeley (1996)

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Coutinho, M.G. (2013). Appendix E: Useful Time Derivatives. In: Guide to Dynamic Simulations of Rigid Bodies and Particle Systems. Simulation Foundations, Methods and Applications. Springer, London. https://doi.org/10.1007/978-1-4471-4417-5_10

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  • DOI: https://doi.org/10.1007/978-1-4471-4417-5_10

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-4416-8

  • Online ISBN: 978-1-4471-4417-5

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