## “An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part one: Single Rigid Bodies”Authors: Pål J. From,
Affiliation: Norwegian University of Life Sciences
Reference: 2012, Vol 33, No 2, pp. 45-60. |

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

**Abstract:**This paper presents the explicit dynamic equations of a mechanical system. The equations are presented so that they can easily be implemented in a simulation software or controller environment and are also well suited for system and controller analysis. The dynamics of a general mechanical system consisting of one or more rigid bodies can be derived from the Lagrangian. We can then use several well known properties of Lie groups to guarantee that these equations are well defined. This will, however, often lead to rather abstract formulation of the dynamic equations that cannot be implemented in a simulation software directly. In this paper we close this gap and show what the explicit dynamic equations look like. These equations can then be implemented directly in a simulation software and no background knowledge on Lie theory and differential geometry on the practitioner´s side is required. This is the first of two papers on this topic. In this paper we derive the dynamics for single rigid bodies, while in the second part we study multibody systems. In addition to making the equations more accessible to practitioners, a motivation behind the papers is to correct a few errors commonly found in literature. For the first time, we show the detailed derivations and how to arrive at the correct set of equations. We also show through some simple examples that these correspond with the classical formulations found from Lagrange´s equations. The dynamics is derived from the Boltzmann--Hamel equations of motion in terms of local position and velocity variables and the mapping to the corresponding quasi-velocities. Finally we present a new theorem which states that the Boltzmann--Hamel formulation of the dynamics is valid for all transformations with a Lie group topology. This has previously only been indicated through examples, but here we also present the formal proof. 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 (423 Kb) DOI: 10.4173/mic.2012.2.2

**DOI forward links to this article:**

[1] Bernard Brogliato (2013), doi:10.1007/s11044-013-9392-5 | |

[2] Pål Johan From (2012), doi:10.4173/mic.2012.2.3 | |

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

[4] Seho Shin, Jonghoon Park and Jaeheung Park (2016), doi:10.1007/s11044-016-9501-3 | |

[5] Juan A. Escalera, Fares J. Abu-Dakka and Mohamed Abderrahim (2016), doi:10.1109/IROS.2016.7759467 | |

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

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

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

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

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

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

**References:**

[1] Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics, Springer-Verlag.

[2] Bullo, F. Lewis, A.D. (2000). Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, Springer Verlag, New York, USA.

[3] Bullo, F. Murray, R. (1999). Tracking for fully actuated mechanical systems: a geometric framework, Automatica, 35(1):17--34 doi:10.1016/S0005-1098(98)00119-8

[4] Duindam, V. (2006). Port-Based Modeling and Control for Efficient Bipedal Walking Robots, Ph.D. thesis, University of Twente.

[5] Duindam, V. Stramigioli, S. (2007). Lagrangian dynamics of open multibody systems with generalized holonomic and nonholonomic joints, In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, San Diego, CA, USA. pp. 3342--3347.

[6] Duindam, V. Stramigioli, S. (2008). Singularity-free dynamic equations of open-chain mechanisms with general holonomic and nonholonomic joints, IEEE Transactions on Robotics, 2.3:517--526 doi:10.1109/TRO.2008.924250

[7] From, P.J., Duindam, V., Pettersen, K.Y., Gravdahl, J.T., Sastry, S. (2010). Singularity-free dynamic equations of vehicle-manipulator systems, Simulation Modelling Practice and Theory, 1.6:712--731 doi:10.1016/j.simpat.2010.01.012

[8] From, P.J., Duindam, V., Stramigioli, S. (2012). Corrections to 'singularity-free dynamic equations of open-chain mechanisms with general holonomic and nonholonomic joints', Submitted to IEEE Transactions on Robotics.

[9] Goldstein, H., Poole, C.P., Safko, J.L. (2001). Classical Mechanics, Addison Wesley, San Francisco, USA.

[10] Murray, R.M., Li, Z., Sastry, S.S. (1994). A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, USA.

[11] Park, F.C., Bobrow, J.E., Ploen, S.R. (1995). A Lie group formulation of robot dynamics, International Journal of Robotics Research, 1.6:609--618 doi:10.1177/027836499501400606

[12] Rossmann, W. (2002). Lie Groups - An introduction through linear algebra, Oxford science publications, Oxford, UK.

[13] Selig, J.M. (2000). Geometric fundamentals of robotics, Springer Verlag, New York, USA doi:10.1142/9789812813282

**BibTeX:**

@article{MIC-2012-2-2,

title={{An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part one: Single Rigid Bodies}},

author={From, Pål J.},

journal={Modeling, Identification and Control},

volume={33},

number={2},

pages={45--60},

year={2012},

doi={10.4173/mic.2012.2.2},

publisher={Norwegian Society of Automatic Control}

};