O(n) Algorithm for Elastic Link/Joint Robots with End-Effector Contact
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This paper deals with the dynamical modeling of flexible multibody systems like elastic robots that go in contact with the environment. Models for elastic systems have a large degree of freedom leading to longer calculation times for solving the equations of motion (EOM). Conventionally, this includes the inversion of the mass matrix with a cubic run time complexity O(n3). By using a subsystem formulation and the Projection Equation an O(n) algorithm can be formulated that significantly reduces the simulation time. Additional contacts with the environment can be included in the equations of motion by the corresponding constraint Jacobian and the contact forces. For the explicit calculation of these forces, normally the inverse of the mass matrix is needed again. A novel algorithm to avoid this inversion is presented. Therein, the contact forces are calculated by additional runs of the same O(n) algorithm that is used without contact. The transition phase between different contact states is treated with the help of Newton’s impact law, again avoiding the inversion of the mass matrix. Simulation results for an elastic robot show the effectiveness of the proposed algorithms.
KeywordsDynamical modeling Elastic multibody systems Ritz approximation O(n) algorithm Contact
This work has been supported by the Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM).
- 3.Brandl, H., Johanni, R., Otter, M.: A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix. In: Proceedings of the IFAC International Symposium on Theory of Robots, Viennam, pp. 365–370 (1986)Google Scholar
- 4.Brandl, H., Johanni, R., Otter, M.: An algorithm for the simulation of multibody systems with kinematic loops. In: Proceedings of 7th IFToMM World Congress on the Theory of Machines and Mechanisms, Sevilla, pp. 407–411 (1987)Google Scholar
- 7.Eigen C++ Library http://eigen.tuxfamily.org
- 10.Gattringer, H., Müller, A.: Dynamic formulations and computational algorithms. In: Goswami, A., Vadakkepat, P. (eds.) Humanoid Robotics: A Reference. Springer, Netherlands (2017)Google Scholar
- 12.Gattringer, H., Müller, A., Springer, K., Jörgl, M.: An efficient method for the dynamical modeling of serial elastic link/joint robots. In: Moreno-Diaz, R., et al. (eds.) Computer Aided Systems Theory – EUROCAST 2015. Series Lecture Notes in Computer Science, vol. 9520, pp. 689–697. Springer, Heidelberg (2015)Google Scholar
- 14.Höbarth, W., Gattringer, H., Bremer, H.: Modeling and control of an articulated robot with flexible links/joints. In: Proceedings of the 9th International Conference on Motion and Vibration Control, Garching (2008)Google Scholar
- 15.Khalil, W.: Dynamic modeling robots using recursive Newton-Euler techniques. In: Filipe, J., Cetto, J., Ferrier, J. (eds.) Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, vol. 2, pp. 19–31. SciTePress, Portugal (2010)Google Scholar