## Efficient formulation of the Gibbs-Appell equations for constrained multibody systems.(English)Zbl 1483.70025

Summary: In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell formulation. Without constructing the Gibbs function, the proposed method provides a minimal set of first-order dynamic equations applicable for multibody constrained systems. Transforming the Newton-Euler equations from physical coordinates to quasivelocity spaces eliminate constraint reaction forces from motion equations. In this study, we develop a general procedure to select effective quasivelocities, which indicate that the proposed quasivelocities satisfy constraints, eliminate Lagrange multipliers, and reduce the number of dynamic equations to degrees of freedom. Besides, we test the validity and efficiency of the proposed approach using three constrained dynamical systems as illustrative examples. Finally, we compare the simulation results with Udwadia-Kalaba, augmented Lagrangian, and other conventional methods.

### MSC:

 7e+56 Dynamics of multibody systems
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### References:

 [1] Ginsberg, J. H., Advanced Engineering Dynamics (1998), Cambridge: Cambridge University Press, Cambridge · Zbl 1103.70300 [2] Bajodah, A. H.; Chen, Y.-H., Canonical generalized inversion form of Kane’s equations of motion for constrained mechanical systems, Nonlinear Systems: Modeling, Estimation, and Stability, 31 (2018) [3] Vilkko, R., Landmark writings in Western mathematics 1640-1940, Rev. Mod. Log., 11, 1-2, 205-218 (2007) [4] Náprstek, J.; Fischer, C., Appell-Gibbs approach in dynamics of non-holonomic systems, Nonlinear Systems-Modeling, Estimation, and Stability (2018), London: IntechOpen, London · Zbl 1299.70042 [5] Desloge, E. A., Relationship between Kane’s equations and the Gibbs-Appell equations, J. Guid. Control Dyn., 10, 1, 120-122 (1987) [6] Borri, M.; Bottasso, C.; Mantegazza, P., Equivalence of Kane’s and Maggi’s equations, Meccanica, 25, 4, 272-274 (1990) · Zbl 0719.70010 [7] Maggi, G. A., Principii della teoria matematica del movimento dei corpi: corso di meccanica razionale (1896), Milano: Ulrico Hoepli, Milano · JFM 26.0777.01 [8] Laulusa, A.; Bauchau, O. A., Review of classical approaches for constraint enforcement in multibody systems, J. Comput. Nonlinear Dyn., 3, 1 (2008) [9] Bajodah, A. H.; Hodges, D. H.; Chen, Y.-H., New form of Kane’s equations of motion for constrained systems, J. Guid. Control Dyn., 26, 1, 79-88 (2003) [10] Pishkenari, H. N.; Yousefsani, S.; Gaskarimahalle, A.; Oskouei, S., A fresh insight into Kane’s equations of motion, Robotica, 35, 3, 498-510 (2017) [11] Udwadia, F. E.; Kalaba, R. E., A new perspective on constrained motion, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 439, 1906, 407-410 (1992) · Zbl 0766.70011 [12] Udwadia, F.; Kalaba, R., The explicit Gibbs-Appell equation and generalized inverse forms, Q. Appl. Math., 56, 2, 277-288 (1998) · Zbl 0958.70010 [13] Blajer, W., A geometrical interpretation and uniform matrix formulation of multibody system dynamics, J. Appl. Math. Mech./Z. Angew. Math. Mech., 81, 4, 247-259 (2001) · Zbl 0984.80014 [14] Pishkenari, H. N.; Heidarzadeh, S., A novel computer-oriented dynamical approach with efficient formulations for multibody systems including ignorable coordinates, Appl. Math. Model., 62, 461-475 (2018) · Zbl 1460.70006 [15] Marques, F.; Souto, A. P.; Flores, P., On the constraints violation in forward dynamics of multibody systems, Multibody Syst. Dyn., 39, 4, 385-419 (2017) · Zbl 1377.70025 [16] Bayo, E.; De Jalon, J. G.; Serna, M. A., A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems, Comput. Methods Appl. Mech. Eng., 71, 2, 183-195 (1988) · Zbl 0666.70021 [17] Bayo, E.; Ledesma, R., Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics, Nonlinear Dyn., 9, 1-2, 113-130 (1996) [18] Dopico, D.; González, F.; Luaces, A.; Saura, M.; García-Vallejo, D., Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections, Nonlinear Dyn., 93, 4, 2039-2056 (2018) [19] Mata, V.; Provenzano, S.; Valero, F.; Cuadrado, J., Serial-robot dynamics algorithms for moderately large numbers of joints, Mech. Mach. Theory, 37, 8, 739-755 (2002) · Zbl 1126.70303 [20] Vossoughi, G.; Pendar, H.; Heidari, Z.; Mohammadi, S., Assisted passive snake-like robots: conception and dynamic modeling using Gibbs-Appell method, Robotica, 26, 3, 267-276 (2008) [21] Korayem, M.; Shafei, A.; Shafei, H., Dynamic modeling of nonholonomic wheeled mobile manipulators with elastic joints using recursive Gibbs-Appell formulation, Sci. Iran., 19, 4, 1092-1104 (2012) [22] Korayem, M.; Shafei, A., Application of recursive Gibbs-Appell formulation in deriving the equations of motion of $$n$$-viscoelastic robotic manipulators in 3D space using Timoshenko beam theory, Acta Astronaut., 83, 273-294 (2013) [23] Korayem, M.; Shafei, A.; Dehkordi, S., Systematic modeling of a chain of $$n$$-flexible link manipulators connected by revolute-prismatic joints using recursive Gibbs-Appell formulation, Arch. Appl. Mech., 84, 2, 187-206 (2014) · Zbl 1298.70004 [24] Marghitu, D. B.; Cojocaru, D., Gibbs-Appell equations of motion for a three link robot with MATLAB, Advances in Robot Design and Intelligent Control, 317-325 (2016), Berlin: Springer, Berlin [25] Malayjerdi, M.; Akbarzadeh, A., Analytical modeling of a 3-d snake robot based on sidewinding locomotion, Int. J. Dyn. Control, 7, 1, 83-93 (2019) [26] Baruh, H., Applied Dynamics (2014), Boca Raton: CRC Press, Boca Raton · Zbl 1418.70002 [27] Gallier, J., The Schur Complement and Symmetric Positive Semidefinite (and Definite) Matrices, 1-12 (2010), University of Pennsylvania: Penn Engineering, University of Pennsylvania [28] Mirtaheri, S.M., Zohoor, H.: Quasi-velocities definition in Lagrangian multibody dynamics. Proc. Inst. Mech. Eng., C J. Mech. Eng. Sci. 0954406221995852 (2021) [29] Udwadia, F. E.; Phohomsiri, P., Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics, Proc. R. Soc. A, Math. Phys. Eng. Sci., 462, 2071, 2097-2117 (2006) · Zbl 1149.70311 [30] Uchida, T.; Vyasarayani, C.; Smart, M.; McPhee, J., Parameter identification for multibody systems expressed in differential-algebraic form, Multibody Syst. Dyn., 31, 4, 393-403 (2014) · Zbl 1303.70016 [31] Zhang, J.; Liu, D.; Liu, Y., A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix, Multibody Syst. Dyn., 36, 1, 87-110 (2016) · Zbl 1372.70028
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