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December  2019, 6(2): 409-427. doi: 10.3934/jcd.2019021

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

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.

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. CelledoniA. 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.

[2]

L. DieciR. 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.

[3]

F. DieleL. 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.

[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.

[5]

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

[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.

[7]

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

[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.

[9]

M. O. HestevikK. O. AustefjordL.-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.

[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.

[11]

A. IserlesM. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

A. A. ShabanaR. 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.

[20]

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

[21]

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

[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.

show all references

References:
[1]

E. CelledoniA. 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.

[2]

L. DieciR. 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.

[3]

F. DieleL. 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.

[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.

[5]

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

[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.

[7]

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

[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.

[9]

M. O. HestevikK. O. AustefjordL.-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.

[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.

[11]

A. IserlesM. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

A. A. ShabanaR. 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.

[20]

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

[21]

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

[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.

Figure 1.  System of chained rigid bodies
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 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 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 5.  4-body pendulum with torsional springs
Figure 6.  16-body
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 8.  Spring loaded/damped chain. The red vector represents $ n $, the unit vector in the direction of the displacement of the spring
Figure 9.  64-body pendulum
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
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
$ 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
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