Lie group integrators for mechanical systems

References

• M. Arnold and O. Brüls, Convergence of the generalized-α scheme for constrained mechanical systems, Multibody Syst. Dyn. 18(2) (2007), pp. 185–202.
   Web of Science ®Google Scholar
• M. Arnold, O. Brüls, and A. Cardona, Error analysis of generalized-α Lie group time integration methods for constrained mechanical systems, Numer. Math. 129(1) (2015), pp. 149–179.
   Web of Science ®Google Scholar
• G. Bogfjellmo and H. Marthinsen, High-order symplectic partitioned Lie group methods, Found. Comput. Math. 16(2) (2015), pp. 493–530.
   Web of Science ®Google Scholar
• O. Bruls and A. Cardona, On the use of lie group time integrators in multibody dynamics, J. Comput. Nonlinear Dyn. 5(3) (2010), p. 031002.
   Web of Science ®Google Scholar
• F. Casas and B. Owren, Cost efficient lie group integrators in the RKMK class, BIT Numer. Math.43(4) (2003), pp. 723–742.
   Web of Science ®Google Scholar
• E. Celledoni, A. Marthinsen, and B. Owren, Commutator-free lie group methods, Future Generation Comput. Syst. 19 (2003), pp. 341–352.
   Web of Science ®Google Scholar
• E. Celledoni, H. Marthinsen, and B. Owren, An introduction to Lie group integrators–basics, new developments and applications, J. Comput. Phys. 257(part B) (2014), pp. 1040–1061.
   Web of Science ®Google Scholar
• E. Celledoni and B. Owren, Lie group methods for rigid body dynamics and time integration on manifolds, Comput. Methods Appl. Mech. Eng. 192(3-4) (2003), pp. 421–438.
   Web of Science ®Google Scholar
• J. Ćesić, V. Jukov, I. Petrovic, and D. Kulić, Full Body Human Motion Estimation on Lie Groups using 3D Marker Position Measurements, Proceedings of the IEEE-RAS International Conference on Humanoid Robotics, Cancun, 15–17 November 2016, 2016.
   Google Scholar
• S.H. Christiansen, H.Z. Munthe-Kaas, and B. Owren, Topics in structure-preserving discretization, Acta Numer. 20 (2011), pp. 1–119.
   Google Scholar
• P.E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci. 3 (1993), pp. 1–33.
   Web of Science ®Google Scholar
• C. Curry and B. Owren, Variable step size commutator free Lie group integrators, Numer. Algorithms82(4) (2019), pp. 1359–1376.
   Web of Science ®Google Scholar
• F. Diele, L. Lopez, and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math. 8(4) (1998), pp. 317–334.
   Web of Science ®Google Scholar
• J.R. Dormand and P.J. Prince, A family of embedded Runge–Kutta formulae, J. Comput. Appl. Math. 6(1) (1980), pp. 19–26.
   Google Scholar
• K. Engø and S. Faltinsen, Numerical integration of Lie–Poisson systems while preserving coadjoint orbits and energy, SIAM J. Numer. Anal. 39(1) (2001), pp. 128–145.
   Web of Science ®Google Scholar
• H. Goldstein, C.P. Poole, and J. Safko, Classical Mechanics, Pearson, Essex, 2013.
   Google Scholar
• V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. Phys. 127 (1980), pp. 220–253.
   Web of Science ®Google Scholar
• E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, Springer, Heidelberg, 2010. Structure-Preserving Algorithms for Ordinary Differential Equations, Reprint of the Second (2006) Edition.
   Google Scholar
• E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin, 1993.
   Google Scholar
• J. Hall and M. Leok, Lie group spectral variational integrators, Found. Comput. Math. 17(1) (2017), pp. 199–257.
   Web of Science ®Google Scholar
• F. Hausdorff, Die symbolische exponentialformel in der gruppentheorie, Leipziger Ber. 58 (1906), pp. 19–48.
   Google Scholar
• H.M. Hilber, T.J.R. Hughes, and R.L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Eng. Structural Dyn. 5(3) (1977), pp. 283–292. cited By 1683.
   Web of Science ®Google Scholar
• D. Holm, Geometric Mechanics: Part II: Rotating, Translating and Rolling, World Scientific Publishing Company, Singapore, 2008.
   Google Scholar
• D. Holm, J. Marsden, and T. Ratiu, The Euler--Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), pp. 1–81.
   Web of Science ®Google Scholar
• S. Holzinger and J. Gerstmayr, Time integration of rigid bodies modelled with three rotation parameters, Multibody Sys Dyn (2021), doi:https://doi.org/10.1007/s11044-021-09778-w
   Web of Science ®Google Scholar
• A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, and A. Zanna, Lie-group methods, Acta Numer. 9 (2000), pp. 215–365.
   Google Scholar
• T. Lee, M. Leok, and N.H. McClamroch, Lie group variational integrators for the full body problem, Comput. Methods Appl. Mech. Eng. 196(29-30) (2007), pp. 2907–2924.
   Web of Science ®Google Scholar
• T. Lee, M. Leok, and N.H. McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds, in A Geometric Approach to Modeling and Analysis, Interaction of Mechanics and Mathematics, Springer, Cham, 2018.
   Google Scholar
• T. Lee, K. Sreenath, and V. Kumar, Geometric Control of Cooperating Multiple Quadrotor UAVs with a Suspended Payload. Proceedings of 52nd IEEE Conference on Decision and Control, Firenze, Italy. IEEE, 2013. pp. 5510–5515.
   Google Scholar
• T. Leitz and S. Leyendecker, Galerkin Lie-group variational integrators based on unit quaternion interpolation, Comput. Methods Appl. Mech. Eng. 338 (2018), pp. 333–361.
   Web of Science ®Google Scholar
• N.E. Leonard and J.E. Marsden, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry, Phys. D 105(1–3) (1997), pp. 130–162.
   Web of Science ®Google Scholar
• D. Lewis and J.C. Simo, Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups, J. Nonlinear Sci. 4 (1994), pp. 253–299.
   Web of Science ®Google Scholar
• A. Lundervold and H.Z. Munthe-Kaas, On Algebraic Structures of Numerical Integration on Vector Spaces and Manifolds, in Faà di Bruno Hopf Algebras, Dyson-Schwinger Equations, and Lie-Butcher Series, volume 21 of IRMA Lect. Math. Theor. Phys., European Mathematical Society Publishing, Zürich, 2015, pp. 219–263.
   Google Scholar
• A. Müller, Coordinate mappings for rigid body motions, ASME J. Comput. Nonlinear Dyn. 12(2) (2017), p. 021010.
   Web of Science ®Google Scholar
• J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York, NY, 1994.
   Google Scholar
• J.E. Marsden, T. Ratiu, and A. Weinstein, Semi-direct products and reduction in mechanics, Trans. Am. Math. Soc. 281 (1884), pp. 147–77.
   Google Scholar
• J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Applied Mathematics, Vol. 17, Springer-Verlag, New York, NY, 1999.
   Google Scholar
• R.H. Merson, An Operational Method for the Study of Integration Processes, Proc. Symp. Data Processing, Weapons Res. Establ. Salisbury, 1957.
   Google Scholar
• J. Moser and A.P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys. 139(2) (1991), pp. 217–243.
   Web of Science ®Google Scholar
• H. Munthe-Kaas, Runge–Kutta methods on Lie groups, BIT 38(1) (1998), pp. 92–111.
   Web of Science ®Google Scholar
• H. Munthe-Kaas, High order Runge–Kutta methods on manifolds, Appl. Numer. Math. 29 (1999), pp. 115–127.
   Web of Science ®Google Scholar
• H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, Phil. Trans. Royal Soc. A 357 (1999), pp. 957–981.
   Web of Science ®Google Scholar
• P.J. Olver, Applications of Lie Groups to Differential Equations, Vol. 107, Springer Science & Business Media, New York, NY, 2000.
   Google Scholar
• B. Owren, Order conditions for commutator-free Lie group methods, J. Phys. A 39(19) (2006), pp. 5585–5599.
   Google Scholar
• B. Owren and A. Marthinsen, Integration methods based on canonical coordinates of the second kind, Numer. Math. 87(4) (2001), pp. 763–790.
   Web of Science ®Google Scholar
• B. Owren, Lie Group Integrators. in Discrete Mechanics, Geometric Integration and Lie-Butcher Series, volume 267 of Springer Proc. Math. Stat., Springer, Cham, 2018, pp. 29–69.
   Google Scholar
• J. Park and W. Chung, Geometric integration on Euclidean group with application to articulated multibody systems, IEEE Trans. Robot. 21(5) (2005), pp. 850–863.
   Web of Science ®Google Scholar
• T. Ratiu, Euler-Poisson equations on Lie algebras and the n-dimensional heavy rigid body, Proc. Nat. Acad. Sci. USA 78(3) (1981), pp. 1327–1328.
   PubMed Web of Science ®Google Scholar
• A. Saccon, Midpoint rule for variational integrators on Lie groups, Int. J. Numer. Methods Eng. 78(11) (2009), pp. 1345–1364.
   Web of Science ®Google Scholar
• J.C. Simo and L. Vu-Quoc, On the dynamics of finite-strain rods undergoing large motions – a geometrically exact approach, Comput. Methods Appl. Mech. Eng. 66 (1988), pp. 125–161.
   Web of Science ®Google Scholar
• A.P. Veselov, Integrable systems with discrete time, and difference operators, Funktsional. Anal. i Prilozhen. 22(2) (1988), pp. 1–13.
   Web of Science ®Google Scholar
• A.M. Vinogradov and B. Kupershmidt, The structure of Hamiltonian mechanics, Funktsional. Anal. i Prilozhen. 32 (1977), pp. 177–243.
   Google Scholar
• F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM 94, Springer-Verlag, New York, NY, 1983.
   Google Scholar
• V. Wieloch and M. Arnold, BDF integrators for constrained mechanical systems on Lie groups, J. Comput. Appl. Math. 387 (2019), p. 112517.
   Web of Science ®Google Scholar