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Comparison of Kane’s and Lagrange’s Methods in Analysis of Constrained Dynamical Systems

Published online by Cambridge University Press:  10 February 2020

Amin Talaeizadeh,
Mahmoodreza Forootan,
Mehdi Zabihi and
Hossein Nejat PishkenariOpen the ORCID record for Hossein Nejat Pishkenari[Opens in a new window]
Amin Talaeizadeh
Affiliation:
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. E-mails: amtalaiezadeh@gmail.com, mahmood.foroutan@yahoo.com, mehdizabihi@yahoo.com
Mahmoodreza Forootan
Affiliation:
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. E-mails: amtalaiezadeh@gmail.com, mahmood.foroutan@yahoo.com, mehdizabihi@yahoo.com
Mehdi Zabihi
Affiliation:
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. E-mails: amtalaiezadeh@gmail.com, mahmood.foroutan@yahoo.com, mehdizabihi@yahoo.com
Hossein Nejat Pishkenari*
Affiliation:
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran. E-mails: amtalaiezadeh@gmail.com, mahmood.foroutan@yahoo.com, mehdizabihi@yahoo.com
*
*Corresponding author. E-mail: nejat@sharif.edu
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Summary

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Dynamic modeling is a fundamental step in analyzing the movement of any mechanical system. Methods for dynamical modeling of constrained systems have been widely developed to improve the accuracy and minimize computational cost during simulations. The necessity to satisfy constraint equations as well as the equations of motion makes it more critical to use numerical techniques that are successful in decreasing the number of computational operations and numerical errors for complex dynamical systems. In this study, performance of a variant of Kane’s method compared to six different techniques based on the Lagrange’s equations is shown. To evaluate the performance of the mentioned methods, snake-like robot dynamics is considered and different aspects such as the number of the most time-consuming computational operations, constraint error, energy error, and CPU time assigned to each method are compared. The simulation results demonstrate the superiority of the variant of Kane’s method concerning the other ones.

Keywords

Kane methodLagrange’s methodSnake robotConstrained dynamical systemsDerivation of motion equations

Information

Type
Articles
Information
Robotica , Volume 38 , Issue 12 , December 2020, pp. 2138 - 2150
Copyright
Copyright © Cambridge University Press 2020

References

Baruh, H., Analytical Dynamics (WCB/McGraw-Hill, Boston, 1999).Google Scholar
Fowles, G. R. and Cassiday, G. L., Analytical Mechanics (Thomson Brooks/Cole, Belmont, CA, 2005).Google Scholar
Kane, T. R., “Dynamics of nonholonomic systems,” J. Appl. Mech. 28(4), 574–578 (1961).CrossRefGoogle Scholar
Kane, T. R. and Levinson, D. A., Dynamics: Theory and Applications (McGraw Hill, New York, 1985).Google Scholar
Roithmayr, C. M. and Hodges, D. H., Dynamics: Theory and Application of Kane’s Method (Cambridge University Press, Cambridge, UK, 2016).CrossRefGoogle Scholar
Willard Gibbs, J., “On the fundamental formulae of dynamics,” American J. Math. 2(1), 49–64 (1879).CrossRefGoogle Scholar
Appell, P., “Sur les mouvements de roulment; equations du mouvement analougues a celles de lagrange,” Comptes Rendus 129, 317–320 (1899).Google Scholar
Papastavridis, J. G., “Maggi’s equations of motion and the determination of constraint reactions,” J. Guidance Control Dyn. 13(2), 213–220 (1990).CrossRefGoogle Scholar
Udwadia, F. E. and Kalaba, R. E., “A new perspective on constrained motion,” Proc. R. Soc. London Ser. A Math. Phys. Sci. 439(1906), 407–410 (1992).Google Scholar
Ginsberg, J., Engineering Dynamics, vol. 10 (Cambridge University Press, Cambridge, UK, 2008).Google Scholar
Khang, N. V., “Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems,” Mech. Res. Commun. 38(4), 294–299 (2011).CrossRefGoogle Scholar
Jarzkebowska, E., Augustynek, K. and Urbaś, A., “Computational Reference Dynamical Model of a Multibody System with First Order Constraints,” ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2017) pp. V006T10A016–V006T10A016.Google Scholar
Marques, F., Souto, A. P. and Flores, P., “On the constraints violation in forward dynamics of multibody systems,” Multi. Syst. Dyn. 39(4), 385–419 (2017).CrossRefGoogle Scholar
Blajer, W., “A geometric unification of constrained system dynamics,” Multi. Syst. Dyn. 1(1), 3–21 (1997).CrossRefGoogle Scholar
Blajer, W., “Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems,” Multi. Syst. Dyn. 7(3), 265–284 (2002).CrossRefGoogle Scholar
Braun, D. J. and Goldfarb, M., “Eliminating constraint drift in the numerical simulation of constrained dynamical systems,” Comput. Methods Appl. Mech. Eng. 198(37–40), 3151–3160 (2009).CrossRefGoogle Scholar
Nejat Pishkenari, H., Yousefsani, S. A., Gaskarimahalle, A. L. and Oskouei, S. B. G., “A fresh insight into Kane’s equations of motion,” Robotica 35(3), 498–510 (2017).CrossRefGoogle Scholar
Bajodah, A. H., Hodges, D. H. and Chen, Y.-H., “Nonminimal generalized Kane’s impulse-momentum relations,” J. Guidance Control Dyn. 27(6), 1088–1092 (2004).CrossRefGoogle Scholar
Peterson, D. L., Gede, G. and Hubbard, M., “Symbolic linearization of equations of motion of constrained multibody systems,” Multi. Syst. Dyn. 33(2), 143–161 (2015).CrossRefGoogle Scholar
Bajodah, A. H., Hodges, D. H. and Chen, Y.-H., “New form of Kane’s equations of motion for constrained systems,” J. Guidance Control Dyn. 26(1), 79–88 (2003).CrossRefGoogle Scholar
Roithmayr, C. M., Bajodah, A. H., Hodges, D. H. and Chen, Y.-H., “Corrigendum: New form of Kane’s equations of motion for constrained systems,” J. Guidance Control Dyn 30(1), 286–288 (2007).CrossRefGoogle Scholar
Bajodah, A. H., Hodges, D. H. and Chen, Y.-H., “Nonminimal Kane’s equations of motion for multibody dynamical systems subject to nonlinear nonholonomic constraints,” Multi. Syst. Dyn. 14(2), 155–187 (2005).CrossRefGoogle Scholar
Roithmayr, C. M. and Hodges, D. H., “Forces associated with non-linear non-holonomic constraint equations,” Int. J. Non-Linear Mech. 45(4), 357–369 (2010).CrossRefGoogle Scholar
Tavakoli Nia, H., Nejat Pishkenari, H. and Meghdari, A., “A recursive approach for the analysis of snake robots using Kane’s equations,” Robotica 24(2), 251–256 (2006).Google Scholar
Raman-Nair, W. and White, M., “A model for deployment of a freefall lifeboat from a moving ramp into waves,” Multi. Syst. Dyn. 29(3), 327–342 (2013).CrossRefGoogle Scholar
Transeth, A. A., Pettersen, K. Y. and Liljebäck, P., “A survey on snake robot modeling and locomotion,” Robotica 27(7), 999–1015 (2009).CrossRefGoogle Scholar
Vossoughi, G., Pendar, H., Heidari, Z. and Mohammadi, S., “Assisted passive snake-like robots: Conception and dynamic modeling using Gibbs–Appell method,” Robotica 26(3), 267–276 (2008).CrossRefGoogle Scholar
Greenwood, D. T., Advanced Dynamics (Cambridge University Press, Cambridge, UK, 2006).Google Scholar