Abstract
This Appendix discusses some of the most common methods used to integrate the differential equations of motion in dynamic simulations. These methods range from simple explicit-Euler, to more sophisticated Runge–Kutta methods, with adaptive time step sizing.
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Notes
- 1.
In this appendix, the word scene refers to the simulated world containing all bodies being simulated.
- 2.
See Appendix D (Chap. 9) for details on how to compute the inertia tensor I b in body-frame coordinates.
- 3.
They may have different constant values for different time intervals, but their value is constant within the same time interval.
- 4.
Stability analysis of this method indicates that the numerical solution is stable for all time-step sizes.
- 5.
The superscript ∗ is used to differentiate the \(\vec{k}_{3}\) time-derivative estimate from the \(\vec{k}_{2}\) estimate, since both refer to the same time t=(t 0+h/2).
References
Baraff, D., Witkin, A.: Physically based modeling. SIGGRAPH Course Notes 13 (1998)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (1996)
Sharp, P.W., Verner, J.H.: Completely embedded Runge–Kutta pairs. SIAM J. Numer. Anal. 31, 1169–1190 (1994)
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Coutinho, M.G. (2013). Appendix B: Numerical Solution of Ordinary Differential Equations of Motion. 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_7
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