American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Study of adaptive symplectic methods for simulating charged particle dynamics
  • JCD Home
  • This Issue
  • Next Article
    Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor
December  2019, 6(2): 409-427. doi: 10.3934/jcd.2019021

Chains of rigid bodies and their numerical simulation by local frame methods

Nicolai Sætran 1, and  Antonella Zanna 2,,

1. 

Nordre Krabbedalen 17, 5178 Loddefjord, Norway
2. 

Matematisk Institutt, Universitetet i Bergen, 5020 Bergen, Norway

* Corresponding author: Antonella Zanna

Received  March 2019 Revised  September 2019 Published  November 2019

We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [18]. In this framework, the dynamics of the $ j $th body is described in a frame relative to the $ (j-1) $th one. Starting from the Lagrangian formulation of the system on $ {{\rm{SO}}}(3)^{N} $, the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.

Keywords: N-body problems, chains of rigid bodies, variations on Lie groups, structure preservation.
Mathematics Subject Classification: Primary: 70F10, 70E55, 65P99; Secondary: 65K99, 65L05.
Citation: Nicolai Sætran, Antonella Zanna. Chains of rigid bodies and their numerical simulation by local frame methods. Journal of Computational Dynamics, 2019, 6 (2) : 409-427. doi: 10.3934/jcd.2019021
References:
[1]

E. Celledoni, A. Marthinsen and B. Owren, Commutator-free Lie group methods, Future Generations Computer Systems, 19 (2003), 341-352.  doi: 10.1016/S0167-739X(02)00161-9.  Google Scholar

[2]

L. Dieci, R. D. Russell and E. S. van Vleck, Unitary integrators and applications to continuous orthonormalization techniques, SIAM J. Num. Anal., 31 (1994), 261-281.  doi: 10.1137/0731014.  Google Scholar

[3]

F. Diele, L. Lopez and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math., 8 (1998), 317-334.  doi: 10.1023/A:1018908700358.  Google Scholar

[4]

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynalics, European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart, 1998, 290 pp. doi: 10.1007/978-3-663-09828-7.  Google Scholar

[5]

R. Featherstone, Rigid Body Dynamics Algorithms, Springer, New York, 2008. doi: 10.1007/978-0-387-74315-8.  Google Scholar

[6]

J. Flatlandsmo, T. Smith, Ø. Halvorsen and T. S. T. Impelluso, Modeling stabilization of crane-induced ship motion with gyroscopic control using the moving frame method, Journal of Computational and Nonlinear Dynamics doi: 10.1115/1.4042323.  Google Scholar

[7]

H. Goldstein, Classical Mechanics, Second edition, Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Reading, Mass., 1980.  Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

M. O. Hestevik, K. O. Austefjord, L.-K. S. Larsen and T. Impelluso, Modelling subsea ROV robotics using the moving frame method, International Journal of Dynamics and Control, 7 (2019), 1306-1320.  doi: 10.1007/s40435-018-0471-6.  Google Scholar

[10]

T. J. Impelluso, The moving frame method in dynamics: Reforming a curriculum and assessment, International Journal of Mechanical Engineering Education, 6 (2018), 158-191.  doi: 10.1177/0306419017730633.  Google Scholar

[11]

A. Iserles, M. P. Calvo and A. Zanna, Runge-Kutta methods for orthogonal and isospectral flows, Appl. Numer. Math., 22 (1996), 153-163.  doi: 10.1016/S0168-9274(96)00029-3.  Google Scholar

[12]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, Cambridge Univ. Press, Cambridge, 9 (2000), 215–365. doi: 10.1017/S0962492900002154.  Google Scholar

[13]

L. O. Jay, Preserving Poisson structure and orthogonality in numerical integration of ordinary differential equations, Comput. Math. Appl., 48 (2004), 237-255.  doi: 10.1016/j.camwa.2003.02.013.  Google Scholar

[14]

M. Leok, L. Taeyoung and N. H. McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds. A Geometric Approach to Modeling and Analysis, Interaction of Mechanics and Mathematics, Springer, Cham, 2018. doi: 10.1007/978-3-319-56953-6.  Google Scholar

[15]

C. Lubich, U. Nowak, U. Pöhle and C. Engstler, MEXX-Numerical Software for the Integration of Constrained Mechanical Multibody Systems, Technical Report Technical Report SC 92–12, ZIB Berlin, 1992. Google Scholar

[16]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[17]

A. Müller, Screw and Lie group theory in multibody kinematics, Multibody Syst. Dyn., 43 (2018), 37-70.  doi: 10.1007/s11044-017-9582-7.  Google Scholar

[18]

H. Murakami and T. J. Impelluso, Moving Frame Method in Dynamics-A Geometrical Approach, Pearson Publishing, 2019. Google Scholar

[19]

A. A. Shabana, R. A. Wehage and Y. L. Hwang, Projection methods in flexible multibody dynamics. Part Ⅱ: Dynamics and recursive projection methods, International Journal for Numerical Methods in Engineering, 35 (1992), 1941-1966.  doi: 10.1002/nme.1620351003.  Google Scholar

[20]

T. Rykkje, Lie Groups and the Principle of Virtual Work Applied to Systems of Linked Rigid Bodies, Master thesis, Universitetet i Bergen, 2018. Google Scholar

[21]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7. Chapman & Hall, London, 1994.  Google Scholar

[22]

R. von Schwering, MultiBody Systems Simulation. Numerical Methods, Algorithms, and Software, Lecture Notes in Computational Science and Engineering, 7. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-58515-9.  Google Scholar

show all references

References:
[1]

E. Celledoni, A. Marthinsen and B. Owren, Commutator-free Lie group methods, Future Generations Computer Systems, 19 (2003), 341-352.  doi: 10.1016/S0167-739X(02)00161-9.  Google Scholar

[2]

L. Dieci, R. D. Russell and E. S. van Vleck, Unitary integrators and applications to continuous orthonormalization techniques, SIAM J. Num. Anal., 31 (1994), 261-281.  doi: 10.1137/0731014.  Google Scholar

[3]

F. Diele, L. Lopez and R. Peluso, The Cayley transform in the numerical solution of unitary differential systems, Adv. Comput. Math., 8 (1998), 317-334.  doi: 10.1023/A:1018908700358.  Google Scholar

[4]

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynalics, European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart, 1998, 290 pp. doi: 10.1007/978-3-663-09828-7.  Google Scholar

[5]

R. Featherstone, Rigid Body Dynamics Algorithms, Springer, New York, 2008. doi: 10.1007/978-0-387-74315-8.  Google Scholar

[6]

J. Flatlandsmo, T. Smith, Ø. Halvorsen and T. S. T. Impelluso, Modeling stabilization of crane-induced ship motion with gyroscopic control using the moving frame method, Journal of Computational and Nonlinear Dynamics doi: 10.1115/1.4042323.  Google Scholar

[7]

H. Goldstein, Classical Mechanics, Second edition, Addison-Wesley Series in Physics, Addison-Wesley Publishing Co., Reading, Mass., 1980.  Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

M. O. Hestevik, K. O. Austefjord, L.-K. S. Larsen and T. Impelluso, Modelling subsea ROV robotics using the moving frame method, International Journal of Dynamics and Control, 7 (2019), 1306-1320.  doi: 10.1007/s40435-018-0471-6.  Google Scholar

[10]

T. J. Impelluso, The moving frame method in dynamics: Reforming a curriculum and assessment, International Journal of Mechanical Engineering Education, 6 (2018), 158-191.  doi: 10.1177/0306419017730633.  Google Scholar

[11]

A. Iserles, M. P. Calvo and A. Zanna, Runge-Kutta methods for orthogonal and isospectral flows, Appl. Numer. Math., 22 (1996), 153-163.  doi: 10.1016/S0168-9274(96)00029-3.  Google Scholar

[12]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, Cambridge Univ. Press, Cambridge, 9 (2000), 215–365. doi: 10.1017/S0962492900002154.  Google Scholar

[13]

L. O. Jay, Preserving Poisson structure and orthogonality in numerical integration of ordinary differential equations, Comput. Math. Appl., 48 (2004), 237-255.  doi: 10.1016/j.camwa.2003.02.013.  Google Scholar

[14]

M. Leok, L. Taeyoung and N. H. McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds. A Geometric Approach to Modeling and Analysis, Interaction of Mechanics and Mathematics, Springer, Cham, 2018. doi: 10.1007/978-3-319-56953-6.  Google Scholar

[15]

C. Lubich, U. Nowak, U. Pöhle and C. Engstler, MEXX-Numerical Software for the Integration of Constrained Mechanical Multibody Systems, Technical Report Technical Report SC 92–12, ZIB Berlin, 1992. Google Scholar

[16]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[17]

A. Müller, Screw and Lie group theory in multibody kinematics, Multibody Syst. Dyn., 43 (2018), 37-70.  doi: 10.1007/s11044-017-9582-7.  Google Scholar

[18]

H. Murakami and T. J. Impelluso, Moving Frame Method in Dynamics-A Geometrical Approach, Pearson Publishing, 2019. Google Scholar

[19]

A. A. Shabana, R. A. Wehage and Y. L. Hwang, Projection methods in flexible multibody dynamics. Part Ⅱ: Dynamics and recursive projection methods, International Journal for Numerical Methods in Engineering, 35 (1992), 1941-1966.  doi: 10.1002/nme.1620351003.  Google Scholar

[20]

T. Rykkje, Lie Groups and the Principle of Virtual Work Applied to Systems of Linked Rigid Bodies, Master thesis, Universitetet i Bergen, 2018. Google Scholar

[21]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7. Chapman & Hall, London, 1994.  Google Scholar

[22]

R. von Schwering, MultiBody Systems Simulation. Numerical Methods, Algorithms, and Software, Lecture Notes in Computational Science and Engineering, 7. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-58515-9.  Google Scholar

Figure 1.  System of chained rigid bodies
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Single solid pendulum, consisting of a rectangular prism. Left: classical planar motion, described by the angle $ \theta $. Right: planar motion, described in 3D by a quaternion and the MFM. At rest (vertical position), the inertial and frame axes are aligned
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Numerical solution for the angle $ \theta $ for the solid pendulum. The MFM is solved with IMR and stepsize $ h = 0.01 $ and the exact solution is computed with RK45 imposing machine precision on the tolerances
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  A simulation of the spinning top, idealized as a rod with a disk (left). Right: position of the center of mass in $ [0,30] $ integrated with step size $ h = 0.01 $. See text for the values of the remaining parameters. The formulation of the equations is identical to that of the single pendulum
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  4-body pendulum with torsional springs
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  16-body
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Conservation of quaternion-unit length (left) and relative energy error (right) for the 16-body pendulum solved with GLRK scheme with a step length $ h = 0.01 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Spring loaded/damped chain. The red vector represents $ n $, the unit vector in the direction of the displacement of the spring
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  64-body pendulum
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Comparison of the error in unit length (for quaternions), relative energy error and function evaulations at $ T = 10 $ for the RK45 and 2-stages GLRK method for a 3D pendulum with 4 links. See text for more details
Link 1 Link 2 Link 3 Link 4 Relative energy error Function evaluations
RK45 5.55e-15 2.47e-14 2.24e-14 2.50e-14 7.22e-14 15794
GL2 2.22e-15 5.55e-16 3.00e-15 1.66e-15 3.00e-13 11334
Table Options

Download as excel

Link 1 Link 2 Link 3 Link 4 Relative energy error Function evaluations
RK45 5.55e-15 2.47e-14 2.24e-14 2.50e-14 7.22e-14 15794
GL2 2.22e-15 5.55e-16 3.00e-15 1.66e-15 3.00e-13 11334
Table 2.  Computational times for systems with $ N $ links. The computational cost grows quadratically with the number of links. See text for more details
$ N $ 4 8 16 32 48 64
Time 719ms$ \pm $4.31ms 2.27s$ \pm $15.8ms 10.7s$ \pm $122ms 45.6$ \pm $1.55s 1m43s$ \pm $1.94s 3m24s$ \pm $5.52s
Table Options

Download as excel

$ N $ 4 8 16 32 48 64
Time 719ms$ \pm $4.31ms 2.27s$ \pm $15.8ms 10.7s$ \pm $122ms 45.6$ \pm $1.55s 1m43s$ \pm $1.94s 3m24s$ \pm $5.52s
[1]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002
[2]

Vadim Azhmyakov, Juan P. Fernández-Gutiérrez, Erik I. Verriest, Stefan W. Pickl. A separation based optimization approach to Dynamic Maximal Covering Location Problems with switched structure. Journal of Industrial & Management Optimization, 2021, 17 (2) : 669-686. doi: 10.3934/jimo.2019128
[3]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306
[4]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307
[5]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2020124
[6]

Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065
[7]

Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146
[8]

Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229
[9]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[10]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118
[11]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
[12]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[13]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[14]

Pablo Neme, Jorge Oviedo. A note on the lattice structure for matching markets via linear programming. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2021001
[15]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123
[16]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117
[17]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322
[18]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[19]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[20]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172

 Impact Factor: 

Tools

Metrics

  • PDF downloads (98)
  • HTML views (338)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences