Abstract
We present an efficient variational integrator for simulating multibody systems. Variational integrators reformulate the equations of motion for multibody systems as discrete Euler-Lagrange (DEL) equation, transforming forward integration into a root-finding problem for the DEL equation. Variational integrators have been shown to be more robust and accurate in preserving fundamental properties of systems, such as momentum and energy, than many frequently used numerical integrators. However, state-of-the-art algorithms suffer from \(O(n^{3})\) complexity, which is prohibitive for articulated multibody systems with a large number of degrees of freedom, n, in generalized coordinates. Our key contribution is to derive a quasi-Newton algorithm that solves the root-finding problem for the DEL equation in O(n), which scales up well for complex multibody systems such as humanoid robots. Our key insight is that the evaluation of DEL equation can be cast into a discrete inverse dynamic problem while the approximation of inverse Jacobian can be cast into a continuous forward dynamic problem. Inspired by Recursive Newton-Euler Algorithm (RNEA) and Articulated Body Algorithm (ABA), we formulate the DEL equation individually for each body rather than for the entire system, such that both inverse and forward dynamic problems can be solved efficiently in O(n). We demonstrate scalability and efficiency of the variational integrator through several case studies.
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References
Marsden, J.E., West, M.: Discrete mechanics and variationalintegrators. Acta Numerica 2001 10 (2001) 357–514
West, M.: Variational integrators. PhD thesis, California Institute of Technology (2004)
Betsch, P., Leyendecker, S.: The discrete null space method for the energy consistent integration of constrained mechanical systems. part ii: Multibody dynamics. International journal for numerical methods in engineering 67(4) (2006) 499–552
Leyendecker, S., Marsden, J.E., Ortiz, M.: Variationalintegrators for constrained dynamical systems. ZAMM-Journal ofApplied Mathematics and Mechanics/ Zeitschrift für AngewandteMathematik und Mechanik 88(9) (2008) 677–708
Leyendecker, S., Ober-Blöbaum, S., Marsden, J.E., Ortiz, M.:Discrete mechanics and optimal control for constrained systems. Optimal Control Applications and Methods 31(6) (2010) 505–528
Johnson, E.R., Murphey, T.D.: Scalable variational integratorsfor constrained mechanical systems in generalized coordinates.Robotics, IEEE Transactions on 25(6) (2009) 1249–1261
Luh, J.Y.,Walker, M.W., Paul, R.P.: On-line computational schemefor mechanical manipulators. Journal of Dynamic Systems, Measurement, and Control 102(2) (1980) 69-76
Featherstone, R.: Rigid body dynamics algorithms. Springer (2014)
Park, F.C., Bobrow, J.E., Ploen, S.R.: A lie group formulation ofrobot dynamics. The International Journal of Robotics Research 14(6) (1995) 609–618
Lee, T.: Computational geometric mechanics and control of rigid bodies. ProQuest (2008)
Kobilarov, M.B., Marsden, J.E.: Discrete geometric optimalcontrol on lie groups. Robotics, IEEE Transactions on 27(4)(2011) 641–655
Kobilarov, M., Crane, K., Desbrun, M.: Lie group integrators for animation and control of vehicles. ACM Transactions on Graphics (TOG) 28(2) (2009)
Murray, R.M., Li, Z., Sastry, S.S.: A mathematical introductionto robotic manipulation. CRC press (1994)
Bou-Rabee, N., Marsden, J.E.: Hamilton.pontryagin integrators on lie groups part i: Introduction and structure-preserving properties. Foundations of Computational Mathematics 9(2) (2009) 197–219
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: structurepreserving algorithms for ordinary di.erential equations. Volume 31. Springer Science & Business Media (2006)
Fan, T., Murphey, T.: Structured linearization of discrete mechanical systems on lie groups: A synthesis of analysis and control. In: 2015 54th IEEE Conference on Decision and Control (CDC), IEEE (2015) 1092–1099
Liu, C.K., Stillman, M., Lee, J., Grey, M.X.: DART - Dynamic Animation and Robotics Toolkit. (2011 (accessed October 30, 2016)) http://dartsim.github.io.
Liu, C.K., Jain, S.: A short tutorial on multibody dynamics. Technical Report GITGVU-15-01-1, Georgia Institute of Technology, School of Interactive Computing (08 2012)
Broyden, C.G.: A class of methods for solving nonlinearsimultaneous equations. Mathematics of computation 19(92)(1965) 577–593
Mirtich, B., Canny, J.: Impulse-based simulation of rigid bodies. In: Proceedings of the 1995 symposium on Interactive 3D graphics, ACM (1995) 181–ff
Kharevych, L.: Geometric interpretation of physical systems for improved elasticity simulations. PhD thesis, Citeseer (2009)
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Lee, J., Liu, C.K., Park, F.C., Srinivasa, S.S. (2020). A Linear-Time Variational Integrator for Multibody Systems. In: Goldberg, K., Abbeel, P., Bekris, K., Miller, L. (eds) Algorithmic Foundations of Robotics XII. Springer Proceedings in Advanced Robotics, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-43089-4_23
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DOI: https://doi.org/10.1007/978-3-030-43089-4_23
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