References:
1.Ginsberg, J.H. “Introduction to analytical mechanics”,In Advanced Engineering Dynamics, F. Padgett, Ed.,2nd Edn., pp. 245-308, Cambridge University Press,Cambridge, UK (1998).https://doi.org/10.1017/CBO9780511800214
2.Dirac, P.A.M. “The Hamilton method”, In Lectures onQuantum Mechanics, Ed., 1st Edn., pp. 1-25, BelforGraduate School of Science, Yeshiva University, NewYork, USA (1964).
3.Capecchi, D. and Drago, A. “On Lagrange’s history ofmechanics”, Meccanica, 40(1), pp. 19-33 (2005).https://doi.org/10.1007/s11012-004-2198-z
4.Köppe, J., Grecksch, W., and Paul, W. “Derivation andapplication of quantum Hamilton equations of motion”, Ann. Phys. (Berl.), 529(3), pp. 1-9 (2017). https://doi.org/10.1002/andp.201600251
5.Borisov, A.V., Mamaev, I.S., and Bizyaev, I.A. “TheJacobi integral in nonholonomic mechanics”, Regul.Chaotic Dyn., 20(3), pp. 383-400 (2015). https://doi.org/10.1134/S1560354715030107
6.Qin, Y., Wang, Z., and Zou, L. “Dynamics of nonlineartransversely vibrating beams: Parametric and closed-form solutions”, Appl. Math. Model., 88, pp. 676-687(2020). https://doi.org/10.1016/j.apm.2020.06.056
7.Papastavridis, J.G. and Yagasaki, K. “Analyticalmechanics, a comprehensive treatise on the dynamics ofconstrained systems; for engineers, physicists, andmathematicians”, ASME. Appl. Mech. Rev., 56(2), pp.B22-B23 (2003). https://doi.org/10.1142/8058
8.Ginsberg, J.H. “Alternative formulations”, InEngineering Dynamics, P. Gordon, Ed., 2nd Edn., pp.552-636, Cambridge University Press, Cambridge, UK(2008). https://doi.org/10.1017/CBO9780511805899
9.Gibbs, J.W. “On the fundamental formulae ofdynamics”, Amer. J. Math., 2(1), pp. 49-64 (1879).https://doi.org/10.2307/2369196
10.Ne_mark, J.I. and Fufaev, N.A. “Analytic dynamics ofnonholonomic systems”, In Dynamics of NonholonomicSystems, J.R. Barbour, Ed., 1st Edn., pp. 87-211,American Mathematical Society (2004).https://doi.org/10.1090/mmono/033
11.Haug, E.J. “Extension of Maggi and Kane equations toholonomic dynamic systems”, J. Comput. Nonlin.Dyn., 13(12), pp. 1-6 (2018). https://doi.org/10.1115/1.4041579
12.Hamel, G., Theoretische Mechanik: Eine EinheitlicheEinführung in die gesamte Mechanik, Springer-Verlag,Berlin (1949).https://doi.org/10.1007/978-3-642-88463-4
13.Kane, T.R. “Dynamics of nonholonomic systems”, J.Appl. Mech., 28(4), pp. 574-578 (1961).https://doi.org/10.1115/1.3641786
14.Kane, T.R. and Levinson, D.A. “Formulation ofequations of motion”, In Dynamics, Theory andApplications, J.P. Holman, Ed., 1st Edn., pp. 190-258,McGraw Hill, New York, USA (1985).
15.Zboiński, K. “Relative kinematics exploited in Kane'sapproach to describe multibody systems in relativemotion”, Acta Mech., 147(1), pp. 19-34 (2001).https://doi.org/10.1007/BF01182349
16.Udwadia, F.E. and Kalaba, R.E. “The fundamentalequation in generalized coordinates”, In AnalyticalDynamics: A New Approach, pp. 175-219, CambridgeUniversity Press, Cambrdige, UK (2007). https://doi.org/10.1017/CBO9780511665479
17.Udwadia, F.E. and Schutte, A.D. “Equations of motionfor general constrained systems in Lagrangianmechanics”, Acta Mech., 213(1), pp. 111-129 (2010).https://doi.org/10.1007/s00707-009-0272-2
18.Zhao, X.M., Chen, Y.H., Zhao, H., et al. “Udwadia–Kalaba equation for constrained mechanical systems:formulation and applications”, Chin. J. Mech.Eng., 31(1), pp. 1-14 (2018).https://doi.org/10.1186/s10033-018-0310-x
19.Liu, P., Hao, Y., and Wang, Q. “Distributed formation-containment control of networked mobile robots usingthe Udwadia–Kalaba approach”, IEEE Trans. NetworkSci. Eng., 11(1), pp. 848-857 (2023).https://doi.org/10.1109/TNSE.2023.3308992
20.Zhen, S., Meng, C., Xiao, H., et al. “Robust approximate constraint following control for SCARA robots’ systemwith uncertainty and experimental validation”, ControlEng. Pract., 138, pp. 1-12 (2023).https://doi.org/10.1016/j.conengprac.2023.105610
21.Straižys, A., Burke, M., and Ramamoorthy, S. “Learning robotic cutting from demonstration: Non-holonomicDMPs using the Udwadia-Kalaba method”, 2023 IEEEInternational Conference on Robotics and Automation(ICRA), London, UK, pp. 5034-5040 (2023). https://doi.org/10.1109/ICRA48891.2023.10160917
22.Zhang, L. and Zhang, D. “A two-loop procedure basedon implicit Runge–Kutta method for index-3 dae ofconstrained dynamic problems”, NonlinearDynam., 85(1), pp. 263-280 (2016).https://doi.org/10.1007/s11071-016-2682-8
23.Marques, F., Souto, A.P., and Flores, P. “On theconstraints violation in forward dynamics of multibodysystems”, Multibody Syst. Dyn., 39(4), pp. 385-419(2017). https://doi.org/10.1007/s11044-016-9530-y
24.Rodríguez, J.I., Jiménez, J.M., Funes, F.J., et al.“Recursive and residual algorithms for the efficientnumerical integration of multi-bodysystems”, Multibody Syst. Dyn., 11(4), pp. 295-320(2004).https://doi.org/10.1023/B:MUBO.0000040798.77064.bc
25.Roberson, R.E. and Schwertassek, R. “Computersimulation”, In Dynamics of Multibody Systems, 1stEdn., pp. 365-411, Springer Berlin, Heidelberg,Germany (1988). https://doi.org/10.1007/978-3-642-86464-3
26.Liu, C.Q. and Huston, R.L. “Another form of equationsof motion for constrained multibodysystems”, Nonlinear Dynam., 51(3), pp. 465-475(2008). https://doi.org/10.1007/s11071-007-9225-2
27.Gattringer, H., Bremer, H. and Kastner, M. “Efficientdynamic modeling for rigid multi-body systems with contact and impact”, Acta Mech., 219(1), pp. 111-128 (2011). https://doi.org/10.1007/s00707-010-0436-0
28.Appell, P. “Sur les mouvements de roulment; equationsdu mouvement analougues a celles deLagrange”, Comptes Rendus., 129, pp. 317-320 (1899).
29.Pius, P. and Selekwa, M. “The equivalence ofBoltzmann–Hamel and Gibbs–Appell equations inmodeling constrained systems”, Int. J. Dyn. Control,11(5), pp. 2101-2111 (2023). https://doi.org/10.1007/s40435-023-01119-3
30.Müller, A. “On the Hamel coefficients and theBoltzmann–Hamel equations for the rigid body”, J.Nonlinear Sci., 31(2), pp. 1-39 (2021).https://doi.org/10.1007/s00332-021-09692-7
31.Mirtaheri, S.M. and Zohoor, H. “Quasi-velocitiesdefinition in Lagrangian multibodydynamics”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical EngineeringScience, 235(20), pp. 4679-4691 (2021). https://doi.org/10.1177/0954406221995852
32.Mirtaheri, S.M. and Zohoor, H. “Efficient formulationof the Gibbs–Appell equations for constrainedmultibody systems”, Multibody Syst. Dyn., 53(3), pp.303-325 (2021). https://doi.org/10.1007/s11044-021-09798-6
33.Honein, T.E. and O’Reilly, O.M. “On the Gibbs–Appellequations for the dynamics of rigid bodies”, J. Appl.Mech., 88(7), pp. 1-8 (2021).https://doi.org/10.1115/1.4051181
34.Mirtaheri, S.M. and Zohoor, H. “The explicit gibbs-appell equations of motion for rigid-body constrainedmechanical system”, 2018 6th RSI InternationalConference on Robotics and Mechatronics(IcRoM), Tehran, Iran, pp. 304-309 (2018).https://doi.org/10.1109/ICRoM.2018.8657637
35.Bajodah, A.H. and Hodges, D.H. “Canonical Kane’sequations of motion for discrete dynamicalsystems”, AIAA J., 57(10), pp. 4226-4240 (2019). https://doi.org/10.2514/1.J057603
36.Korayem, M.H., Shafei, A.M., and Shafei, H.R.“Dynamic modeling of nonholonomic wheeled mobilemanipulators with elastic joints using recursive Gibbs–Appell formulation”, Sci. Iran., 19(4), pp. 1092-1104(2012). https://doi.org/10.1016/j.scient.2012.05.001
37.Talaeizadeh, A., Forootan, M., Zabihi, M., et al.“Comparison of Kane’s and Lagrange’s methods inanalysis of constrained dynamical systems”, Robotica,38(12), pp. 2138-2150 (2020).https://doi.org/10.1017/S0263574719001899
38.Korayem, M.H. and Shafei A.M. “Motion equationsproper for forward dynamics of robotic manipulatorwith flexible links by using recursive Gibbs-Appellformulation”, Sci. Iran., 16(6), pp. 479-495 (2009).
39.Malayjerdi, M. and Akbarzadeh, A. “Analyticalmodeling of a 3-d snake robot based on sidewindinglocomotion”, Int. J. Dyn. Control, 7(1), pp. 83–93(2019). https://doi.org/10.1007/s40435-018-0441-z
40.Bilgili, D., Budak, E., and Altintas, Y. “Multibodydynamic modeling of five-axis machine tools withimproved efficiency”, Mech. Syst. Signal Pr., 171, pp. 1-20 (2022). https://doi.org/10.1016/j.ymssp.2022.108945
41.Pishkenari, H.N. and Heidarzadeh, S. “A novelcomputer-oriented dynamical approach with efficientformulations for multibody systems including ignorablecoordinates”, Appl. Math. Model., 62, pp. 461-475(2018).https://doi.org/10.1016/j.apm.2018.06.012
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