### “An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part Two: Multibody Systems”

**Authors:**Pål J. From,

**Affiliation:**Norwegian University of Life Sciences

**Reference:**2012, Vol 33, No 2, pp. 61-68.

**Keywords:**Lagrangian mechanics, singularities, implementation, Lie theory

**Abstract:**This paper presents the explicit dynamic equations of multibody mechanical systems. This is the second paper on this topic. In the first paper the dynamics of a single rigid body from the Boltzmann--Hamel equations were derived. In this paper these results are extended to also include multibody systems. We show that when quasi-velocities are used, the part of the dynamic equations that appear from the partial derivatives of the system kinematics are identical to the single rigid body case, but in addition we get terms that come from the partial derivatives of the inertia matrix, which are not present in the single rigid body case. We present for the first time the complete and correct derivation of multibody systems based on the Boltzmann--Hamel formulation of the dynamics in Lagrangian form where local position and velocity variables are used in the derivation to obtain the singularity-free dynamic equations. The final equations are written in global variables for both position and velocity. The main motivation of these papers is to allow practitioners not familiar with differential geometry to implement the dynamic equations of rigid bodies without the presence of singularities. Presenting the explicit dynamic equations also allows for more insight into the dynamic structure of the system. Another motivation is to correct some errors commonly found in the literature. Unfortunately, the formulation of the Boltzmann-Hamel equations used here are presented incorrectly. This has been corrected by the authors, but we present here, for the first time, the detailed mathematical details on how to arrive at the correct equations. We also show through examples that using the equations presented here, the dynamics of a single rigid body is reduced to the standard equations on a Lagrangian form, for example Euler´s equations for rotational motion and Euler--Lagrange equations for free motion.

PDF (306 Kb) DOI: 10.4173/mic.2012.2.3

**DOI forward links to this article:**

[1] Pål Johan From, Vincent Duindam and Stefano Stramigioli (2012), doi:10.1109/TRO.2012.2206853 |

[2] Pål Johan From, Jan Tommy Gravdahl and Kristin Ytterstad Pettersen (2014), doi:10.1007/978-1-4471-5463-1_3 |

[3] Pål Johan From, Jan Tommy Gravdahl and Kristin Ytterstad Pettersen (2014), doi:10.1007/978-1-4471-5463-1_5 |

[4] Pål Johan From, Jan Tommy Gravdahl and Kristin Ytterstad Pettersen (2014), doi:10.1007/978-1-4471-5463-1_6 |

[5] Pål Johan From, Jan Tommy Gravdahl and Kristin Ytterstad Pettersen (2014), doi:10.1007/978-1-4471-5463-1_8 |

[6] Pål Johan From, Jan Tommy Gravdahl and Kristin Ytterstad Pettersen (2014), doi:10.1007/978-1-4471-5463-1_7 |

[7] Balint Varga, Selina Meier, Stefan Schwab and Soren Hohmann (2019), doi:10.1109/ICMECH.2019.8722886 |

[8] Jake Welde and Vijay Kumar (2020), doi:10.1109/ICRA40945.2020.9196705 |

[9] Jake Welde, James Paulos and Vijay Kumar (2021), doi:10.1109/LRA.2021.3051572 |

[10] Kaimeng Wang and Hehua Ju (2022), doi:10.1016/j.ijnonlinmec.2022.104033 |

[11] Kaimeng Wang, Hehua Ju and Yang yang (2022), doi:10.1016/j.apm.2022.05.008 |

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**BibTeX:**

@article{MIC-2012-2-3,

title={{An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part Two: Multibody Systems}},

author={From, Pål J.},

journal={Modeling, Identification and Control},

volume={33},

number={2},

pages={61--68},

year={2012},

doi={10.4173/mic.2012.2.3},

publisher={Norwegian Society of Automatic Control}

};