Publication: Rigid body with rotors and reduction by stages
Full text at PDC
Advisors (or tutors)
World Scientific Publishing
Rigid body with rotors is a widespread mechanical system modeled after the direct product SO(3)×S 1×S 1×S 1, which under mild assumptions is the symmetry group of the system. In this paper, the authors present and compare different Lagrangian reduction procedures: Euler-Poincaré reduction by the whole group and reduction by stages in different orders or using different connections. The exposition keeps track of the equivalence of equations as well as corresponding conservation laws.
 M.A. Berbel, M. Castrillón López, Reduction (by stages) in the whole Lagrange Poincaré category arXiv:1912.10763  M.A. Berbel, M. Castrillón López, Lagrangian Reduction by Stages in Field Theory arXiv:2007.14854  A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, T.S. Ratiu, Dissipation Induced Instabilities, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 11 (1994), 7–90.  H. Cendra, J.E. Marsden, S. Pekarsky, T.S. Ratiu, Variational principles for Lie-Poisson and Hamilton-Poincaré equations, Moscow Mathematical Journal 3 (2003), 833–867.  H. Cendra, J.E. Marsden, T.S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc. 152 (2001), no. 722.  A.V. Doroshin, Homoclinic solutions and motion chaotization in attitude dynamics of a multi-spin spacecraft, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 7, 2528–2552.  S. Gajbhiye, R. N. Banavar, Geometric tracking control for a nonholonomic system: a spherical robot Internat. J. Robust Nonlinear Control 26 11 (2016), 2436–2454.  P.S. Krishnaprasad, C.A. Berenstein, On the equilibria of rigid spacecraft with rotors, Systems Control Lett. 4 (1984), no. 3, 157–163.  J.E. Marsden, J. Scheurle, The reduced Euler-Lagrange equations, Fields Institute Comm., 1 139–164, 1993.  A. Nayak, R. N. Banavar, D. Maithripala, Almost-global tracking for a rigid body with internal rotors Eur. J. Control 42 (2018), 59–66.  M. Puta, R. Tudoran, R. Tudoran, Poisson manifolds and Bermejo-Fairén construction of Casimirs, Tensor (N.S.) 66 (2005), no. 1, 59–70.  M. Svinin, A. Morinaga, M. Yamamoto, On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors Regular and Chaotic Dynamics,18(1-2) (2013), 126–143.